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Beyond Advanced · Coordinate Geometry

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1
Fixed points are A(0,0)A(0,0), B(6,0)B(6,0) and C(0,4)C(0,4). A point PP moves in the plane such that PA=2 PBPA=2\,PB. Find the minimum possible value of PA+2 PCPA+2\,PC. Give the answer correct to 2 decimal places.
Not attempted
xyABC
Your answer: —
Correct answer: 14.42
2
A triangle ABCABC has circumcentre at the origin OO and orthocentre H(1,4)H(1,4). One vertex is A(5,0)A(5,0). Find the area of triangle ABCABC (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 26.24
3
Triangle ABCABC has vertices A(0,0)A(0,0), B(14,0)B(14,0) and C(5,12)C(5,12). Let O,I,HO,I,H and NN denote its circumcentre, incentre, orthocentre and the centre of its nine-point circle, and let R,rR,r be its circumradius and inradius. Which of the following statements are correct?
Not attempted
xyABC
A
OI2=6564OI^2=\frac{65}{64}
B
The excentre opposite the vertex AA is the point (21,14)(21,14).
C
II lies on the nine-point circle of ABCABC.
D
The incircle and the nine-point circle touch each other internally.
4
A line segment ABAB of fixed length 1010 slides with AA on the positive xx-axis and BB on the positive yy-axis (both strictly positive). Let II be the incentre of triangle OABOAB, where OO is the origin. The locus of II is a part of a straight line. Find the length of this part (correct to 2 decimal places).
Not attempted
xyABO
Your answer: —
Correct answer: 2.93
5
Let S\mathcal{S} be the set of points in the plane for which the sum of the perpendicular distances from the lines y=xy=x and y=−xy=-x is at most 424\sqrt2. Find the area of the part of S\mathcal{S} lying inside the circle x2+y2=25x^2+y^2=25 (correct to 2 decimal places).
Not attempted
xyy = xy = −x
Your answer: —
Correct answer: 62.19
6
Let A(−2,0)A(-2,0) and B(4,0)B(4,0) be fixed points and let PP be a variable point. Which of the following statements are correct?
Not attempted
A
The locus of PP for PA2+PB2=40PA^2+PB^2=40 is a circle of radius 11\sqrt{11}.
B
The locus of PP for PA2−PB2=12PA^2-PB^2=12 is the line x=3x=3.
C
The locus of PP for PA=2 PBPA=2\,PB is a circle of radius 44.
D
The locus of PP for ∣PA−PB∣=6|PA-PB|=6 consists of two rays, one starting at AA and the other at BB.
7
Triangle ABCABC has vertices A(1,2)A(1,2), B(5,−2)B(5,-2) and C(0,6)C(0,6). As the point PP varies over the line x+y=10x+y=10, the sum PA2+PB2+PC2PA^2+PB^2+PC^2 takes a least value. Find it.
Not attempted
xyABCx + y = 10
Your answer: —
Correct answer: 100
8
In triangle ABCABC with A(0,0)A(0,0), B(14,0)B(14,0) and C(5,12)C(5,12), the bisector of the exterior angle at AA meets the line BCBC at EE. Find the length AEAE.
Not attempted
xyABC
A
561356 \sqrt{13}
B
28139\frac{28 \sqrt{13}}{9}
C
41497\sqrt{41497}
D
210210
9
Triangle ABCABC has vertices A(2,12)A\left(2,\frac12\right), B(−1,−1)B(-1,-1) and C(4,14)C\left(4,\frac14\right). Its orthocentre is the point
Not attempted
A
(−18, −8)\left(-\frac18,\,-8\right)
B
(18, 8)\left(\frac18,\,8\right)
C
(8, 18)\left(8,\,\frac18\right)
D
(18, −8)\left(\frac18,\,-8\right)
10
Triangle ABCABC has vertices A(0,0)A(0,0), B(21,0)B(21,0) and C(6,8)C(6,8). Let O,I,G,H,NO,I,G,H,N be its circumcentre, incentre, centroid, orthocentre and nine-point centre. Match each squared distance in List-I with its value in List-II. List-IList-IIP)OI21)246564Q)IG22)16936R)ON23)11905256S)HG24)119051445)1190564\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} OI^2 & \text{1)} \frac{2465}{64} \\ \text{Q)} IG^2 & \text{2)} \frac{169}{36} \\ \text{R)} ON^2 & \text{3)} \frac{11905}{256} \\ \text{S)} HG^2 & \text{4)} \frac{11905}{144} \\ & \text{5)} \frac{11905}{64} \end{array}
Not attempted
xyABC
A
P-1, Q-2, R-4, S-3
B
P-2, Q-1, R-3, S-4
C
P-1, Q-2, R-3, S-4
D
P-1, Q-5, R-3, S-2
11
A line through the point P(1,8)P(1,8) meets the positive xx-axis at AA and the positive yy-axis at BB. Find the least possible length of the segment ABAB (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 11.18
12
Points A(1,1)A(1,1) and B(5,3)B(5,3) are given. A point PP moves along the xx-axis. Find the greatest value, in degrees, of the angle ∠APB\angle APB (correct to 2 decimal places).
Not attempted
xyABx
Your answer: —
Correct answer: 97.24
13
Let P=(3,4)P=(3,4), O=(0,0)O=(0,0) and A=(6,0)A=(6,0). Which of the following statements are correct?
Not attempted
A
Exactly one straight line passes through PP at distance 44 from OO.
B
Exactly one straight line passes through PP at distance 55 from OO.
C
Exactly one straight line passes through PP and is equidistant from OO and AA.
D
Exactly two straight lines pass through PP at distance 22 from OO and at distance 33 from AA.
14
A ray of light starts at A(7,2)A(7,2), strikes the xx-axis, is reflected, then strikes the line y=xy=x, is reflected again and finally passes through B(4,3)B(4,3). Find the total length of the path from AA to BB (correct to 2 decimal places).
Not attempted
xyy = xAB
Your answer: —
Correct answer: 7.21
15
Find the number of points with integer coordinates lying strictly inside the triangle bounded by the coordinate axes and the line 3x+5y=603x+5y=60.
Not attempted
Your answer: —
Correct answer: 103
16
The lines y=xy=x and y=3xy=3x meet at the origin OO. A line through P(4,5)P(4,5) cuts them at AA and BB respectively. Find the minimum possible area of triangle OABOAB.
Not attempted
xyPy = xy = 3x
Your answer: —
Correct answer: 7
17
A triangle has its three sides lying along lines of slopes 11, 22 and 33 (the triangle can be placed anywhere in the plane). Which of the following statements are correct?
Not attempted
A
The triangle is obtuse-angled.
B
Its sides are in the ratio 1:5:101:\sqrt5:\sqrt{10}.
C
Its orthocentre lies inside the triangle.
D
The longest side lies along the line of slope 22.
18
Through the point P(1,2)P(1,2) a straight line is drawn so that the segment of it lying between the lines 2x+y−1=02x+y-1=0 and x−y+5=0x-y+5=0 is bisected at PP. The equation of this line is
Not attempted
xyPx − y + 5 = 02x + y − 1 = 0
A
5x+7y=195x+7y=19
xyP
B
x+2y=5x+2y=5
xyP
C
x−2y=−3x-2y=-3
xyP
D
11x+y=1311x+y=13
xyP
19
Through the point P(2,1)P(2,1), two lines are drawn that meet the line x+y=11x+y=11 at points at a distance 66 from PP. The sum of the slopes of these two lines is
Not attempted
A
187\frac{18}{7}
B
718\frac{7}{18}
C
22
D
−187-\frac{18}{7}
20
All lines considered pass through the point (2,3)(2,3) and have non-zero intercepts on both axes. For each condition in List-I, match the number of such lines from List-II. List-IList-IIP)The line and the axes enclose a triangle of area 61)2Q)The line and the axes enclose a triangle of area 122)3R)The line and the axes enclose a triangle of area 243)4S)The line and the axes enclose a triangle of area 6 and both intercepts are positive4)15)0\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The line and the axes enclose a triangle of area } 6 & \text{1)} 2 \\ \text{Q)} \text{The line and the axes enclose a triangle of area } 12 & \text{2)} 3 \\ \text{R)} \text{The line and the axes enclose a triangle of area } 24 & \text{3)} 4 \\ \text{S)} \text{The line and the axes enclose a triangle of area } 6 \text{ and both intercepts are positive} & \text{4)} 1 \\ & \text{5)} 0 \end{array}
Not attempted
A
P-1, Q-2, R-4, S-5
B
P-1, Q-2, R-3, S-5
C
P-2, Q-3, R-1, S-5
D
P-1, Q-3, R-2, S-4
21
The line x+2y−3=0x+2y-3=0 is reflected in the mirror line 3x−y−2=03x-y-2=0. The equation of the reflected line (the image of the whole line) is
Not attempted
xyx + 2y − 3 = 0mirror 3x − y − 2 = 0
A
5x+10y−8=05x+10y-8=0
xy
B
x+3y−4=0x+3y-4=0
xy
C
2x+11y−13=02x+11y-13=0
xy
D
x−2y+1=0x-2y+1=0
xy
22
Points A(0,0)A(0,0) and B(8,2)B(8,2) are given. A point PP moves along the line x+y=6x+y=6. Find the maximum value of ∣PA−PB∣|PA-PB| (correct to 2 decimal places).
Not attempted
xyABx + y = 6
Your answer: —
Correct answer: 4.47
23
The point (t, t2−4)(t,\,t^2-4) lies strictly inside the triangle with vertices (−2,−1)(-2,-1), (6,−1)(6,-1) and (−2,7)(-2,7) exactly when tt belongs to
Not attempted
xy
A
(3,37−12)\left(\sqrt3,\frac{\sqrt{37}-1}{2}\right)
B
(−2,37−12)\left(-2,\frac{\sqrt{37}-1}{2}\right)
C
(−1+372,−3)∪(3,37−12)\left(-\frac{1+\sqrt{37}}{2},-\sqrt3\right)\cup\left(\sqrt3,\frac{\sqrt{37}-1}{2}\right)
D
(−2,−3)∪(3,37−12)\left(-2,-\sqrt3\right)\cup\left(\sqrt3,\frac{\sqrt{37}-1}{2}\right)
24
In triangle ABCABC with A(0,0)A(0,0), B(14,0)B(14,0) and C(5,12)C(5,12), an interior point PP has perpendicular distances to the sides ABAB, CACA and BCBC in the ratio 1:2:31:2:3. Find the distance from PP to BCBC (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 5.93
25
A line ℓ\ell through the fixed point A(6,2)A(6,2) rotates about AA. Let MM and NN be the feet of the perpendiculars to ℓ\ell from the origin OO and from B(−3,5)B(-3,5) respectively. Find the maximum possible length of MNMN (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 5.83
26
Consider the lines x−2y+4=0x-2y+4=0 and 2x−y+3=02x-y+3=0. Which of the following statements are correct?
Not attempted
xyOx − 2y + 4 = 02x − y + 3 = 0
A
The bisector of the angle between the lines that contains the origin is x+y=1x+y=1.
B
The angle between the lines that contains the origin is acute.
C
The point (1,1)(1,1) lies inside an acute angle formed by the lines.
D
The bisector of the acute angle between the lines has slope 11.
27
The point P(5,2)P(5,2) is reflected successively in the line y=xy=x, then in the line y=3 xy=\sqrt3\,x, and finally in the line y=0y=0. Find the xx-coordinate of the final image (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 3.33
28
The four sides of a square, taken in order, pass through the points A(0,0)A(0,0), B(5,1)B(5,1), C(6,5)C(6,5) and D(1,4)D(1,4) respectively (the first side through AA, the second through BB, the third through CC, the fourth through DD). More than one such square exists. Find the sum of the areas of all such squares (correct to 2 decimal places).
Not attempted
xyABCD
Your answer: —
Correct answer: 8.6
29
Consider the family of lines (1+k)x+(2k−1)y+(3−k)=0(1+k)x+(2k-1)y+(3-k)=0, k∈Rk\in\mathbb{R}, and the point Q(−23,103)Q\left(-\frac23,\frac{10}{3}\right). Which of the following statements are correct?
Not attempted
A
Every member of the family passes through the same fixed point.
B
The distance from QQ to a member of the family attains a greatest value, equal to 5\sqrt5.
C
The distance from QQ to a member of the family can take every value in [0,5)[0,\sqrt5).
D
The line x+2y=1x+2y=1 is a member of the family.
30
Each row of List-I gives a pair of lines. Match it with the equation in List-II of the bisector of the acute angle between them. List-IList-IIP)3x+4y+5=0 and 5x−12y+13=01)1x+8y=0Q)3x+4y+5=0 and 5x+12y+13=02)32x+56y+65=0R)3x+4y+5=0 and −5x+12y+13=03)7x+56y+65=0S)3x+4y+5=0 and 12x+5y+13=04)99x+77y+130=05)32x−4y+65=0\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} 3x+4y+5=0 \text{ and } 5x-12y+13=0 & \text{1)} 1x+8y=0 \\ \text{Q)} 3x+4y+5=0 \text{ and } 5x+12y+13=0 & \text{2)} 32x+56y+65=0 \\ \text{R)} 3x+4y+5=0 \text{ and } -5x+12y+13=0 & \text{3)} 7x+56y+65=0 \\ \text{S)} 3x+4y+5=0 \text{ and } 12x+5y+13=0 & \text{4)} 99x+77y+130=0 \\ & \text{5)} 32x-4y+65=0 \end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-5, Q-2, R-4, S-3
C
P-1, Q-3, R-2, S-4
D
P-2, Q-1, R-3, S-4
31
The numbers a,b,ca,b,c are non-zero reals in harmonic progression. Consider all lines xa+yb+1c=0\frac xa+\frac yb+\frac1c=0 obtainable in this way. How many of these lines are at distance exactly 33 from the point (4,−2)(4,-2)?
Not attempted
A
00
B
11
C
22
D
infinitely many
32
The chord of the circle x2+y2−4x−6y+c=0x^2+y^2-4x-6y+c=0 cut off by the line x+y=1x+y=1 subtends a right angle at the origin OO. If AA and BB are the end points of this chord, find the area of triangle OABOAB (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 0.71
33
Let a,b,ca,b,c be real numbers and consider the line ℓ: (a−b)x+(b−c)y+(c−a)=0\ell:\ (a-b)x+(b-c)y+(c-a)=0 (not all of a−ba-b, b−cb-c zero). Which of the following statements are correct?
Not attempted
A
Every such line passes through the point (1,1)(1,1).
B
If a,b,ca,b,c are in arithmetic progression with non-zero common difference, then ℓ\ell is always the same line x+y=1x+y=1.
C
If a,b,ca,b,c are in geometric progression with common ratio 22, then ℓ\ell is the line x+2y=1x+2y=1.
D
No line of this family can be the line x=yx=y.
34
The equation 2x2+7xy+3y2+8x+14y+λ=02x^2+7xy+3y^2+8x+14y+\lambda=0 represents a pair of straight lines. Find the area of the triangle bounded by these two lines and the line y=3y=3 (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 11.25
35
In triangle ABCABC with A(0,0)A(0,0), B(8,0)B(8,0) and C(2,6)C(2,6), the point DD on BCBC satisfies BD:DC=1:2BD:DC=1:2, the point EE on CACA satisfies CE:EA=2:3CE:EA=2:3, and FF is a point on ABAB such that the lines ADAD, BEBE and CFCF are concurrent. The point of concurrency is
Not attempted
xyABCDE
A
(103,2)\left(\frac{10}{3},2\right)
B
(5411,1811)\left(\frac{54}{11},\frac{18}{11}\right)
C
(4811,1811)\left(\frac{48}{11},\frac{18}{11}\right)
D
(5411,1211)\left(\frac{54}{11},\frac{12}{11}\right)
36
A variable line through the fixed point (2,3)(2,3) meets the xx-axis at AA and the yy-axis at BB. The midpoint MM of ABAB traces a curve. The eccentricity of this curve is
Not attempted
A
22
B
3\sqrt3
C
2\sqrt2
D
23\frac{2}{\sqrt3}
37
Find the sum of the perpendicular distances of the point (4,3)(4,3) from the two lines represented by 3x2+8xy−3y2=03x^2+8xy-3y^2=0 (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 6.96
38
For each integer kk with −6≤k≤6-6\le k\le6 consider the three lines 3x−y−2=03x-y-2=0, x+y−2=0x+y-2=0 and (k−2)x+(k+2)y−(k2−2k+3)=0(k-2)x+(k+2)y-(k^2-2k+3)=0. For how many of these integers kk do the three lines form a triangle (three distinct vertices)?
Not attempted
Your answer: —
Correct answer: 10
39
Match each homogeneous equation in List-I, which represents a pair of straight lines through the origin, with the acute angle between those lines in List-II. List-IList-IIP)x2−4xy+y2=01)0∘Q)2x2−5xy−2y2=02)60∘R)3x2−10xy+3y2=03)90∘S)x2−23 xy+3y2=04)tan⁡−1435)45∘\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^2-4xy+y^2=0 & \text{1)} 0^\circ \\ \text{Q)} 2x^2-5xy-2y^2=0 & \text{2)} 60^\circ \\ \text{R)} 3x^2-10xy+3y^2=0 & \text{3)} 90^\circ \\ \text{S)} x^2-2\sqrt3\,xy+3y^2=0 & \text{4)} \tan^{-1}\frac43 \\ & \text{5)} 45^\circ \end{array}
Not attempted
A
P-2, Q-3, R-1, S-4
B
P-3, Q-2, R-4, S-1
C
P-2, Q-5, R-4, S-1
D
P-2, Q-3, R-4, S-1
40
Consider the pair of lines 3x2+8xy−3y2=03x^2+8xy-3y^2=0. Which of the following statements are correct?
Not attempted
A
The two lines make equal angles with the yy-axis.
B
The combined equation of the bisectors of the angles between the lines is 2x2−3xy−2y2=02x^2-3xy-2y^2=0.
C
The two lines and the line x+y=7x+y=7 enclose a triangle of area 2458\frac{245}{8}.
D
The pair of lines through the origin perpendicular to these two lines is represented by the same equation.
41
The pair of lines x2−5xy+6y2=0x^2-5xy+6y^2=0 is translated so that its point of intersection moves to P(1,2)P(1,2). The area of the triangle enclosed by the translated lines and the yy-axis is
Not attempted
A
112\frac{1}{12}
B
16\frac{1}{6}
C
124\frac{1}{24}
D
512\frac{5}{12}
42
Two circles pass through the points A(1,2)A(1,2) and B(3,4)B(3,4) and touch the xx-axis. Find the area of the quadrilateral whose vertices are the centres of the two circles and their two points of contact with the xx-axis.
Not attempted
xyAB
Your answer: —
Correct answer: 48
43
Let S1: x2+y2−4x−2y+1=0S_1:\ x^2+y^2-4x-2y+1=0 and S2: x2+y2−2x−4y+1=0S_2:\ x^2+y^2-2x-4y+1=0. For real λ\lambda let F\mathcal{F} denote the set of circles S1+λS2=0S_1+\lambda S_2=0 (taking only those values of λ\lambda for which the equation is a real circle). Which of the following statements are correct?
Not attempted
A
The smallest circle in F\mathcal{F} has radius 142\frac{\sqrt{14}}{2}.
B
The member with λ=−1\lambda=-1 is the circle having the common chord of S1S_1 and S2S_2 as a diameter.
C
F\mathcal{F} contains a circle of zero radius.
D
Exactly one member of F\mathcal{F} touches the xx-axis, although S2S_2 itself also touches it.
44
The circle x2+y2−8x−6y+16=0x^2+y^2-8x-6y+16=0 is cut by the line x+2y−6=0x+2y-6=0 at two points. Find the radius of the smallest circle that passes through both these points (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 2.41
45
A variable circle passes through the origin and cuts off algebraic intercepts aa on the xx-axis and bb on the yy-axis (the signed lengths from the origin to its other points of intersection with the axes), where a+b=8a+b=8. Find the least possible radius (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 2.83
46
Circles of different radii touch both coordinate axes and pass through the point (3,6)(3,6). The distance between their centres is
Not attempted
xy(3, 6)
A
1212
B
12212\sqrt2
C
626\sqrt2
D
18218\sqrt2
47
Chords of the circle x2+y2=25x^2+y^2=25 are drawn through the point P(3,4)P(3,4) on it. The locus of the midpoint of such a chord is the circle
Not attempted
A
x2+y2−6x−8y=0x^2+y^2-6x-8y=0
xyP
B
x2+y2−3x−4y=25x^2+y^2-3x-4y=25
xyP
C
x2+y2−3x−4y=0x^2+y^2-3x-4y=0
xyP
D
x2+y2−3x2−2y=0x^2+y^2-\frac{3x}{2}-2y=0
xyP
48
Consider the circle x2+y2−2x+4y−4=0x^2+y^2-2x+4y-4=0. Which of the following statements are correct?
Not attempted
A
The maximum value of 3x+4y3x+4y on the circle is 1515.
B
The minimum value of x2+y2x^2+y^2 over points of the circle is 14−6514-6\sqrt5.
C
The ratio yx\frac yx for points of the circle has a greatest value.
D
The circle has exactly four points with integer coordinates.
49
The point P(5,5)P(5,5) lies inside the circle x2+y2−4x−6y−12=0x^2+y^2-4x-6y-12=0. The longest and the shortest chords of the circle through PP are drawn. Find the area of the quadrilateral whose vertices are the four end points of these two chords (correct to 2 decimal places).
Not attempted
xyCP
Your answer: —
Correct answer: 34.64
50
Match each circle described in List-I with its radius in List-II. List-IList-IIP)The circle passing through (0,0), (6,0) and (0,8)1)1Q)The circle with centre (3,0) that cuts the circle x2+y2=4 orthogonally2)13R)The circle having the points (−1,2) and (3,−4) as ends of a diameter3)5S)The circle touching the x-axis at (1,0) and passing through (0,1)4)55)10\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The circle passing through } (0,0) \text{, } (6,0) \text{ and } (0,8) & \text{1)} 1 \\ \text{Q)} \text{The circle with centre } (3,0) \text{ that cuts the circle } x^2+y^2=4 \text{ orthogonally} & \text{2)} \sqrt{13} \\ \text{R)} \text{The circle having the points } (-1,2) \text{ and } (3,-4) \text{ as ends of a diameter} & \text{3)} 5 \\ \text{S)} \text{The circle touching the } x \text{-axis at } (1,0) \text{ and passing through } (0,1) & \text{4)} \sqrt5 \\ & \text{5)} \sqrt{10} \end{array}
Not attempted
A
P-3, Q-4, R-1, S-2
B
P-4, Q-3, R-2, S-1
C
P-3, Q-5, R-2, S-1
D
P-3, Q-4, R-2, S-1
51
The circumcircle of the triangle bounded by the lines y=xy=x, y=2xy=2x and x+y=3x+y=3 has the equation
Not attempted
xyy = xy = 2xx + y = 3
A
x2+y2−x−2y=0x^2+y^2-x-2y=0
xy
B
x2+y2−2x−y=0x^2+y^2-2x-y=0
xy
C
x2+y2−3x−3y=0x^2+y^2-3x-3y=0
xy
D
x2+y2−x−2y+1=0x^2+y^2-x-2y+1=0
xy
52
Tangents are drawn from the point P(10,0)P(10,0) to the circle x2+y2=25x^2+y^2=25, touching it at AA and BB. The area of triangle PABPAB is
Not attempted
xyPO
A
25325\sqrt3
B
7534\frac{75\sqrt3}{4}
C
2532\frac{25\sqrt3}{2}
D
12534\frac{125\sqrt3}{4}
53
Tangents are drawn from an external point PP to the circle x2+y2=16x^2+y^2=16, and ABAB is the chord of contact. Which of the following statements are correct?
Not attempted
A
If PP moves on the line x+2y=8x+2y=8, then the chord of contact ABAB always passes through the fixed point (2,4)(2,4).
B
If PP moves on the circle x2+y2=64x^2+y^2=64, then ABAB always touches the circle x2+y2=1x^2+y^2=1.
C
For P=(8,0)P=(8,0) the two tangents are perpendicular.
D
For P=(8,0)P=(8,0) the length of ABAB is 434\sqrt3.
54
A point QQ moves on the line 3x+4y=293x+4y=29. The two tangents from QQ to the circle (x−1)2+(y−2)2=9(x-1)^2+(y-2)^2=9 are perpendicular for exactly two positions of QQ. Find the distance between these two positions (correct to 2 decimal places).
Not attempted
xyC3x + 4y = 29
Your answer: —
Correct answer: 4.49
55
A circle touches the line y=xy=x at the point (1,1)(1,1) and passes through the point (4,0)(4,0). Its radius is
Not attempted
xyTBy = x
A
52\frac52
B
54\frac{5}{4}
C
522\frac{5}{2\sqrt2}
D
524\frac{5\sqrt2}{4}
56
Tangents are drawn from a point QQ on the line 3x+4y=253x+4y=25 to the circle x2+y2=9x^2+y^2=9. Which of the following statements are correct?
Not attempted
A
The least possible length of a tangent from such a point is 44.
B
The least length occurs at the point (4,3)(4,3).
C
Tangents of length exactly 55 are obtained from two points of the line, each at distance 33 from the foot of the perpendicular from the origin.
D
For the point where the tangent length is least, the triangle formed by the two tangents and the chord of contact has area 6.46.4.
57
The point P(12,0)P(12,0) lies inside the circle x2+y2=169x^2+y^2=169. A line through PP meets the circle at AA and BB (with PA=4PA=4 and PP between AA and BB) and meets the polar of PP with respect to the circle at QQ, with AA between PP and QQ. Find the length PQPQ (correct to 2 decimal places).
Not attempted
xyPO
Your answer: —
Correct answer: 22.22
58
For the circle S: x2+y2−4x−6y−12=0S:\ x^2+y^2-4x-6y-12=0, match each quantity in List-I with its value in List-II. List-IList-IIP)The length of the tangent from the point (8,3) to S1)11Q)The distance of the origin from the tangent to S at the point (5,7)2)435R)The radius of the circle that is the locus of points from which perpendicular tangents can be drawn to S3)52S)The length of the chord of contact of the tangents from (8,3)4)51135)10\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The length of the tangent from the point } (8,3) \text{ to } S & \text{1)} \sqrt{11} \\ \text{Q)} \text{The distance of the origin from the tangent to } S \text{ at the point } (5,7) & \text{2)} \frac{43}{5} \\ \text{R)} \text{The radius of the circle that is the locus of points from which perpendicular tangents can be drawn to } S & \text{3)} 5\sqrt2 \\ \text{S)} \text{The length of the chord of contact of the tangents from } (8,3) & \text{4)} \frac{5\sqrt{11}}{3} \\ & \text{5)} 10 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-4, S-3
C
P-1, Q-3, R-2, S-4
D
P-1, Q-2, R-3, S-4
59
The tangents drawn from a point QQ on the line 3x+4y=203x+4y=20 to the circle x2+y2=9x^2+y^2=9 include an angle of at least 60∘60^\circ. The length of the portion of the line made up of such points QQ is
Not attempted
xyO3x + 4y = 20
A
454\sqrt5
B
636\sqrt3
C
222\sqrt2
D
828\sqrt2
60
The tangents at the end points of any chord of the circle x2+y2=25x^2+y^2=25 that passes through the fixed point (1,2)(1,2) meet at a point TT. The locus of TT is the line
Not attempted
A
x+2y=5x+2y=5
xy
B
x+2y=25x+2y=25
xy
C
2x+y=252x+y=25
xy
D
x+2y=252x+2y=\frac{25}{2}
xy
61
Consider the circle (x−1)2+(y−2)2=4(x-1)^2+(y-2)^2=4. Which of the following statements are correct?
Not attempted
A
Exactly two tangents can be drawn from (1,−3)(1,-3), and neither is vertical.
B
One of the tangents drawn from (3,4)(3,4) is a vertical line.
C
There are exactly two real values of kk for which the line y=2x+ky=2x+k touches the circle.
D
The length of the tangent from (5,2)(5,2) is 232\sqrt3.
62
For each integer kk with −10≤k≤10-10\le k\le10, let Γk\Gamma_k be the circle of radius 33 centred at (k, k+1)(k,\,k+1). For how many such kk does Γk\Gamma_k intersect the circle x2+y2=25x^2+y^2=25 in exactly two points?
Not attempted
xycentres on y = x + 1
Your answer: —
Correct answer: 10
63
Circle Γ1\Gamma_1 has centre (0,0)(0,0) and radius 22, and circle Γ2\Gamma_2 has centre (6,8)(6,8) and radius 33. Which of the following statements are correct?
Not attempted
xyC₁C₂
A
Each direct common tangent has length 3113\sqrt{11} between its points of contact.
B
Each transverse common tangent has length 535\sqrt3 between its points of contact.
C
The transverse common tangents meet at (245,325)\left(\frac{24}{5},\frac{32}{5}\right).
D
The direct common tangents meet at (−6,−8)(-6,-8).
64
Three circles have equations x2+y2=1x^2+y^2=1, (x−8)2+y2=4(x-8)^2+y^2=4 and (x−4)2+(y−7)2=4(x-4)^2+(y-7)^2=4. A fourth circle cuts all three circles orthogonally. Its radius is
Not attempted
A
6916\frac{69}{16}
B
6116\frac{61}{16}
C
4761256\frac{4761}{256}
D
94\frac{9}{4}
65
Two circles x2+y2=1x^2+y^2=1 and (x−6)2+y2=4(x-6)^2+y^2=4 are given. A variable circle cuts both of them orthogonally. Find the least possible radius of such a circle (correct to 2 decimal places).
Not attempted
xyC₁C₂
Your answer: —
Correct answer: 2.56
66
The circles x2+y2=36x^2+y^2=36 and (x−10)2+y2=64(x-10)^2+y^2=64 intersect. The length of their common chord is
Not attempted
A
245\frac{24}{5}
B
485\frac{48}{5}
C
1010
D
365\frac{36}{5}
67
Two circles of radii 44 and 99 touch each other externally and both touch the xx-axis from above. A third circle, lying above the xx-axis in the gap between them, touches both circles and the xx-axis. Find its radius (correct to 2 decimal places).
Not attempted
xy
Your answer: —
Correct answer: 1.44
68
Consider the circles S1: x2+y2=4S_1:\ x^2+y^2=4 and S2: (x−3)2+y2=1S_2:\ (x-3)^2+y^2=1. Which of the following statements are correct?
Not attempted
A
The radical axis of S1S_1 and S2S_2 is the line x=52x=\frac52.
B
The circle (x−2)2+(y−4)2=16(x-2)^2+(y-4)^2=16 cuts both S1S_1 and S2S_2 orthogonally.
C
The radical axis of S1S_1 and S2S_2 is the line joining their centres.
D
The radical axis of S1S_1 and S2S_2 is their common tangent at the point where they touch.
69
For the circles x2+y2=4x^2+y^2=4 and (x−6)2+y2=1(x-6)^2+y^2=1, the acute angle between the two transverse (internal) common tangents is
Not attempted
xyC₁C₂
A
30∘30^\circ
B
90∘90^\circ
C
60∘60^\circ
D
2sin⁡−1162\sin^{-1}\frac16
70
The circles x2+y2=4x^2+y^2=4 and x2+y2−4x=0x^2+y^2-4x=0 intersect at two points. The circle passing through both these points and touching the line x=6x=6 (other than the second circle itself, which does not touch it) has the equation
Not attempted
A
11x2+11y2−32x−12=011x^2+11y^2-32x-12=0
xy
B
5x2+5y2−32x−12=05x^2+5y^2-32x-12=0
xy
C
x2+y2−6x+4=0x^2+y^2-6x+4=0
xy
D
5x2+5y2−32x+12=05x^2+5y^2-32x+12=0
xy
71
The circle Γ: x2+y2−4x+6y−12=0\Gamma:\ x^2+y^2-4x+6y-12=0 is given. Each equation in List-I is a second circle. Match it with the number of common tangents it has with Γ\Gamma in List-II. List-IList-IIP)x2+y2−12x+32=01)0Q)x2+y2−18x+80=02)1R)x2+y2−4x+6y+4=03)2S)x2+y2−10x−2y−74=04)35)4\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^2+y^2-12x+32=0 & \text{1)} 0 \\ \text{Q)} x^2+y^2-18x+80=0 & \text{2)} 1 \\ \text{R)} x^2+y^2-4x+6y+4=0 & \text{3)} 2 \\ \text{S)} x^2+y^2-10x-2y-74=0 & \text{4)} 3 \\ & \text{5)} 4 \end{array}
Not attempted
A
P-3, Q-5, R-1, S-2
B
P-3, Q-5, R-2, S-1
C
P-5, Q-3, R-1, S-2
D
P-3, Q-1, R-5, S-2
72
A variable circle cuts the circle x2+y2=1x^2+y^2=1 orthogonally and touches the line x=2x=2. The locus of its centre is a parabola whose focus is the point
Not attempted
A
(54,0)\left(\frac54,0\right)
B
(14,0)\left(\frac14,0\right)
C
(−34,0)\left(-\frac34,0\right)
D
(12,0)\left(\frac12,0\right)
73
A focal chord of the parabola y2=8xy^2=8x is divided by the focus in the ratio 1:31:3. The length of the chord is
Not attempted
xySx = −2
A
163\frac{16}{3}
B
643\frac{64}{3}
C
323\frac{32}{3}
D
1212
74
A focal chord PQPQ of the parabola y2=16xy^2=16x has length 2525. Find the area of triangle OPQOPQ, where OO is the vertex.
Not attempted
Your answer: —
Correct answer: 40
75
Let PQPQ be a variable focal chord of the parabola y2=4xy^2=4x with focus SS. Which of the following statements are correct?
Not attempted
A
The circle having PQPQ as a diameter always touches the directrix x=−1x=-1.
B
The midpoint of PQPQ lies on the parabola y2=4(x−1)y^2=4(x-1).
C
If PP and QQ have parameters t1,t2t_1,t_2 (points (t2,2t)(t^2,2t)), then t1+t2=0t_1+t_2=0.
D
The length of the shortest focal chord is 22.
76
For the parabola y2=12xy^2=12x with vertex OO, let PQPQ be a variable focal chord. The centroid of triangle OPQOPQ traces a parabola. Its latus rectum has length
Not attempted
A
1212
B
33
C
66
D
44
77
Let d1d_1 and d2d_2 be the least distances from the points (3,0)(3,0) and (10,0)(10,0) respectively to points of the parabola y2=8xy^2=8x. Find d1+d2d_1+d_2.
Not attempted
Your answer: —
Correct answer: 11
78
Consider the curve y=2x2−8x+5y=2x^2-8x+5. Which of the following statements are correct?
Not attempted
A
Its vertex is the point (−2,−3)(-2,-3).
B
Its focus is the point (2,−238)\left(2,-\frac{23}{8}\right).
C
Its directrix is the line y=−258y=-\frac{25}{8}.
D
Its latus rectum has length 22.
79
An equilateral triangle is inscribed in the parabola y2=8xy^2=8x with one vertex at the vertex of the parabola. The length of its side is
Not attempted
A
16316\sqrt3
B
838\sqrt3
C
32332\sqrt3
D
12312\sqrt3
80
Each equation in List-I represents a parabola. Match it with its focus in List-II. List-IList-IIP)y2−4y−8x+20=01)(4,2)Q)x2+6x+8y+1=02)(−3,−1)R)y2+6y+4x−3=03)(2,−3)S)x2−4x−12y−8=04)(2,2)5)(3,−3)\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} y^2-4y-8x+20=0 & \text{1)} (4,2) \\ \text{Q)} x^2+6x+8y+1=0 & \text{2)} (-3,-1) \\ \text{R)} y^2+6y+4x-3=0 & \text{3)} (2,-3) \\ \text{S)} x^2-4x-12y-8=0 & \text{4)} (2,2) \\ & \text{5)} (3,-3) \end{array}
Not attempted
A
P-1, Q-2, R-4, S-3
B
P-1, Q-2, R-3, S-4
C
P-2, Q-1, R-3, S-4
D
P-4, Q-2, R-3, S-1
81
The circle passing through the vertex of the parabola y2=8xy^2=8x and through the end points of its latus rectum has radius
Not attempted
xyO
A
44
B
252\sqrt5
C
55
D
66
82
Three points of the parabola y2=12xy^2=12x have parameters t=3,4,5t=3,4,5 in the representation (3t2,6t)(3t^2,6t). The area of the triangle formed by these points is
Not attempted
A
99
B
3636
C
5454
D
1818
83
Chords PQPQ of the parabola y2=4xy^2=4x are drawn such that ∠POQ=90∘\angle POQ=90^\circ, where OO is the vertex. The least possible length of such a chord is
Not attempted
A
88
B
44
C
454\sqrt5
D
828\sqrt2
84
Tangents are drawn from the point P(0,4)P(0,4) to the parabola y2=4xy^2=4x, touching it at AA and BB. Find the area of triangle PABPAB.
Not attempted
xyPO
Your answer: —
Correct answer: 32
85
Consider normals to the parabola y2=4xy^2=4x, where the point (t2,2t)(t^2,2t) is called the point tt. Which of the following statements are correct?
Not attempted
A
Exactly one normal to the parabola passes through (11,6)(11,6).
B
For the three normals through (11,6)(11,6), the sum of the ordinates of their feet is 00.
C
Exactly three distinct normals pass through (3,2)(3,2).
D
The normal at the point t=1t=1 meets the parabola again at the point (9,6)(9,6).
86
A ray of light travelling parallel to the axis of the parabola y2=8xy^2=8x comes along the line y=6y=6 from the right (from the line x=10x=10), strikes the parabola (the concave side), passes through the focus, strikes the parabola again and leaves parallel to the axis. Find the total length of the path of the light from the line x=10x=10 (where it enters along y=6y=6) back to the line x=10x=10 (where it leaves),
Not attempted
xySy = 6x = 10
Your answer: —
Correct answer: 24
87
A focal chord of the parabola y2=8xy^2=8x has slope 22. The tangents at its end points meet at the point
Not attempted
A
(−2,4)(-2,4)
B
(−2,2)(-2,2)
C
(−2,1)(-2,1)
D
(2,−2)(2,-2)
88
Tangents are drawn to the parabola y2=8xy^2=8x at the three points (2,4)(2,4), (8,8)(8,8) and (18,12)(18,12). The area of the triangle formed by these three tangents is
Not attempted
xy
A
88
B
22
C
44
D
1616
89
Let P=(4,4)P=(4,4) be a point of the parabola y2=4xy^2=4x with focus S(1,0)S(1,0). Which of the following statements are correct?
Not attempted
A
The foot of the perpendicular from SS to the tangent at PP lies on the tangent at the vertex.
B
The tangent at PP meets the axis at (−4,0)(-4,0), and the tangent at the vertex passes through the midpoint of this intercept (between PP and that point).
C
The normal at PP passes through the focus.
D
The tangent at PP makes equal angles with SPSP and with the line through PP parallel to the axis.
90
The parabola y2=8xy^2=8x and the circle x2+y2=2x^2+y^2=2 have two common tangents. The area of the quadrilateral whose vertices are the four points of contact is
Not attempted
xy
A
1212
B
1818
C
99
D
1515
91
For the parabola y2=8xy^2=8x, match each statement in List-I with the corresponding number in List-II. List-IList-IIP)The number of tangents that can be drawn from the point (−1,1)1)0Q)The number of tangents that can be drawn from the point (1,1)2)1R)The number of normals that can be drawn from the point (3,0)3)2S)The number of normals that can be drawn from the point (12,0)4)35)4\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The number of tangents that can be drawn from the point } (-1,1) & \text{1)} 0 \\ \text{Q)} \text{The number of tangents that can be drawn from the point } (1,1) & \text{2)} 1 \\ \text{R)} \text{The number of normals that can be drawn from the point } (3,0) & \text{3)} 2 \\ \text{S)} \text{The number of normals that can be drawn from the point } (12,0) & \text{4)} 3 \\ & \text{5)} 4 \end{array}
Not attempted
A
P-3, Q-1, R-2, S-4
B
P-3, Q-1, R-4, S-2
C
P-1, Q-3, R-2, S-4
D
P-3, Q-2, R-1, S-4
92
Perpendicular tangents are drawn to the parabola y2=8xy^2=8x from points lying on the circle (x−1)2+y2=25(x-1)^2+y^2=25. The distance between the two such points is
Not attempted
A
66
B
88
C
1010
D
828\sqrt2
93
The tangent and the normal at the point P(2,4)P(2,4) of the parabola y2=8xy^2=8x meet the axis of the parabola at TT and GG respectively. The area of triangle PTGPTG is
Not attempted
xyP
A
88
B
3232
C
1616
D
2424
94
Two tangents are drawn from the point (−2,3)(-2,3) to the parabola y2=8xy^2=8x. Which of the following statements are correct?
Not attempted
A
The two tangents are perpendicular.
B
One of the tangents is 2y=x+82y=x+8.
C
The chord of contact is 3y=4(x−2)3y=4(x-2).
D
The points of contact are (8,8)(8,8) and (12,−2)\left(\frac12,-2\right).
95
For the ellipse x225+y216=1\frac{x^2}{25}+\frac{y^2}{16}=1 with foci SS and S′S', how many points PP of the ellipse satisfy SP⋅S′P=17SP\cdot S'P=17?
Not attempted
xySS'
A
44
B
22
C
88
D
00
96
Let SS and S′S' be the foci of the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1. A point PP of the ellipse satisfies ∠SPS′=90∘\angle SPS'=90^\circ. Find the area of triangle SPS′SPS'.
Not attempted
xySS'
Your answer: —
Correct answer: 9
97
Consider the ellipse 4x2+9y2=364x^2+9y^2=36. Which of the following statements are correct?
Not attempted
A
Its eccentricity is 53\frac{\sqrt5}{3}.
B
The length of its latus rectum is 43\frac43.
C
Its directrices are the lines x=±95x=\pm\frac{9}{\sqrt5}.
D
Its auxiliary circle has radius 22.
98
A point PP moves in the plane so that PA+PB=10PA+PB=10, where A=(1,2)A=(1,2) and B=(5,2)B=(5,2). The equation of the locus of PP is
Not attempted
A
(x−3)225+(y−2)216=1\frac{(x-3)^2}{25}+\frac{(y-2)^2}{16}=1
xyAB
B
(x−3)225+(y−2)221=1\frac{(x-3)^2}{25}+\frac{(y-2)^2}{21}=1
xyAB
C
(x−3)221+(y−2)225=1\frac{(x-3)^2}{21}+\frac{(y-2)^2}{25}=1
xyAB
D
(x−1)225+(y−2)221=1\frac{(x-1)^2}{25}+\frac{(y-2)^2}{21}=1
xyAB
99
A circle with centre PP touches externally the circle with centre A(−3,0)A(-3,0) and radius 11, and touches internally the circle with centre B(3,0)B(3,0) and radius 99. As the circle varies, the length of the latus rectum of the locus of PP is
Not attempted
xyAB
A
165\frac{16}{5}
B
185\frac{18}{5}
C
325\frac{32}{5}
D
88
100
Find the maximum value of 3x+4y3x+4y for points (x,y)(x,y) of the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1 (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 12.04
101
Consider the ellipse x216+y27=1\frac{x^2}{16}+\frac{y^2}{7}=1 with foci SS and S′S'. Which of the following statements are correct?
Not attempted
A
For the point of the ellipse with abscissa 22, the focal distances are 52\frac52 and 112\frac{11}2.
B
The shortest focal chord has length 72\frac72.
C
For the point with abscissa 22 in the first quadrant, SP2+S′P2=36SP^2+S'P^2=36.
D
Exactly four points of the ellipse are at distance 52\frac52 from the focus S(3,0)S(3,0).
102
Match each ellipse in List-I with its eccentricity in List-II. List-IList-IIP)9x2+25y2=2251)35Q)4x2+y2=42)45R)16x2+25y2−64x+50y−311=03)32S)x2+2y2=84)125)12\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} 9x^2+25y^2=225 & \text{1)} \frac35 \\ \text{Q)} 4x^2+y^2=4 & \text{2)} \frac45 \\ \text{R)} 16x^2+25y^2-64x+50y-311=0 & \text{3)} \frac{\sqrt3}{2} \\ \text{S)} x^2+2y^2=8 & \text{4)} \frac12 \\ & \text{5)} \frac1{\sqrt2} \end{array}
Not attempted
A
P-2, Q-3, R-5, S-1
B
P-1, Q-3, R-2, S-5
C
P-2, Q-4, R-1, S-5
D
P-2, Q-3, R-1, S-5
103
The tangents at the end points of the latera recta of the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 form a quadrilateral. Its area is
Not attempted
xySS'
A
1252\frac{125}{2}
B
1254\frac{125}{4}
C
2254\frac{225}{4}
D
5050
104
For the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1, let PP and QQ be the points with eccentric angles θ\theta and θ+π2\theta+\frac{\pi}{2}, and OO the centre. For θ=π7\theta=\frac{\pi}{7}, find the area of triangle OPQOPQ (correct to 2 decimal places).
Not attempted
Your answer: —
Correct answer: 7.5
105
A point QQ moves along the line x+y=4x+y=4. The two tangents from QQ to the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1 are perpendicular for exactly two positions of QQ. Find the distance between these two positions (correct to 2 decimal places).
Not attempted
xyx + y = 4
Your answer: —
Correct answer: 4.47
106
Consider the ellipse x216+y29=1\frac{x^2}{16}+\frac{y^2}{9}=1 with foci (±7,0)(\pm\sqrt7,0). Which of the following statements are correct?
Not attempted
A
The tangents of slope 11 are y=x±7y=x\pm\sqrt7.
B
For the tangent y=x+5y=x+5, the product of the perpendicular distances of the two foci from it is 99.
C
For the tangent y=x+5y=x+5, the foot of the perpendicular from the focus (7,0)(\sqrt7,0) lies on the circle x2+y2=16x^2+y^2=16.
D
The tangent of slope 11 with positive intercept passes through the point (0,4)(0,4).
107
The normal at a variable point of the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 meets the major and minor axes at GG and gg respectively. The locus of the midpoint of GgGg is
Not attempted
A
25x2+9y2=1625x^2+9y^2=16
B
25x2+9y2=6425x^2+9y^2=64
C
9x2+25y2=649x^2+25y^2=64
D
x2+y2=16x^2+y^2=16
108
A variable tangent to the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 cuts off a segment between the coordinate axes. The least possible length of this segment is
Not attempted
A
34\sqrt{34}
B
2152\sqrt{15}
C
88
D
1515
109
Tangents are drawn from the point (6,0)(6,0) to the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1, touching it at AA and BB. The area of triangle formed by the two tangents and the chord ABAB is
Not attempted
xyP
A
939\sqrt3
B
934\frac{9\sqrt3}{4}
C
272\frac{27}{2}
D
932\frac{9\sqrt3}{2}
110
A variable tangent to the ellipse x216+y29=1\frac{x^2}{16}+\frac{y^2}{9}=1 meets the xx-axis at TT and the yy-axis at UU. The locus of the midpoint of TUTU is the curve
Not attempted
A
4x2+94y2=1\frac{4}{x^2}+\frac{9}{4y^2}=1
B
16x2+9y2=1\frac{16}{x^2}+\frac{9}{y^2}=1
C
x24+4y29=1\frac{x^2}{4}+\frac{4y^2}{9}=1
D
4x2+9y2=1\frac{4}{x^2}+\frac{9}{y^2}=1
111
For the ellipse x216+y29=1\frac{x^2}{16}+\frac{y^2}{9}=1, consider the chord whose midpoint is (2,1)(2,1). Which of the following statements are correct?
Not attempted
A
The chord is the line 9x+8y=189x+8y=18.
B
The midpoints of all chords parallel to this chord lie on the line x−2y=0x-2y=0.
C
The tangents at the ends of this chord meet at the point (7213,3613)\left(\frac{72}{13},\frac{36}{13}\right).
D
The point (2,1)(2,1) lies inside the ellipse.
112
Let PP and P′P' be the ends of a diameter of the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1, and Q,Q′Q,Q' the ends of the conjugate diameter (the diameter bisecting all chords parallel to PP′PP'). The tangents at P,P′,Q,Q′P,P',Q,Q' form a parallelogram. Find its area.
Not attempted
Your answer: —
Correct answer: 60
113
For the ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1, match the point with the given eccentric angle θ\theta (List-I) with the equation of the tangent at that point (List-II). List-IList-IIP)θ=01)x=5Q)θ=π22)y=3R)θ=π43)3x+5y=152S)θ=π34)3x+53 y=305)x+y=5\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \theta=0 & \text{1)} x=5 \\ \text{Q)} \theta=\frac\pi2 & \text{2)} y=3 \\ \text{R)} \theta=\frac\pi4 & \text{3)} 3x+5y=15\sqrt2 \\ \text{S)} \theta=\frac\pi3 & \text{4)} 3x+5\sqrt3\,y=30 \\ & \text{5)} x+y=5 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-3, S-4
C
P-1, Q-2, R-4, S-3
D
P-1, Q-5, R-3, S-4
114
A ray of light starts from the focus S(3,0)S(3,0) of the ellipse x225+y216=1\frac{x^2}{25}+\frac{y^2}{16}=1, is reflected from the ellipse, passes through the other focus S′S', is reflected again and returns to SS. The total length of the path is
Not attempted
xySS'
A
1010
B
1616
C
2020
D
3030
115
The ellipse x225+y29=1\frac{x^2}{25}+\frac{y^2}{9}=1 and the circle x2+y2=16x^2+y^2=16 have four common tangents. They enclose a quadrilateral whose area is
Not attempted
xy
A
2563\frac{256}{3}
B
25637\frac{256}{3\sqrt7}
C
5129\frac{512}{9}
D
51237\frac{512}{3\sqrt7}
116
For each integer λ\lambda with −10≤λ≤10-10\le\lambda\le10 consider the equation 2x2+λxy+3y2−4x+6y+5=02x^2+\lambda xy+3y^2-4x+6y+5=0. For how many values of λ\lambda does it represent a real, non-degenerate ellipse?
Not attempted
Your answer: —
Correct answer: 4
117
Find the area enclosed by the curve 5x2+4xy+8y2=365x^2+4xy+8y^2=36 (correct to 2 decimal places).
Not attempted
xy
Your answer: —
Correct answer: 18.85
118
Consider the curve x2−2xy+y2−8x−8y=0x^2-2xy+y^2-8x-8y=0. Which of the following statements are correct?
Not attempted
xy
A
It is a parabola.
B
Its focus is the point (2,2)(2,2).
C
Its directrix is the line x+y=2x+y=2.
D
The length of its latus rectum is 88.
119
The curve 3x2+4xy+3y2−2x−2y−1=03x^2+4xy+3y^2-2x-2y-1=0 is an ellipse. Its eccentricity is
Not attempted
xy
A
25\frac{2}{\sqrt5}
B
15\frac{1}{\sqrt5}
C
25\sqrt{\frac25}
D
45\frac{4}{5}
120
Find the distance between the foci of the hyperbola x2+4xy−2y2=6x^2+4xy-2y^2=6 (correct to 2 decimal places).
Not attempted
xy
Your answer: —
Correct answer: 4.47
121
The asymptotes of the hyperbola 2x2+5xy+2y2+4x+5y+3=02x^2+5xy+2y^2+4x+5y+3=0 are the lines
Not attempted
A
2x+y+1=0 and x+2y+2=02x+y+1=0\ \text{and}\ x+2y+2=0
xy
B
2x+y+2=0 and x+2y+1=02x+y+2=0\ \text{and}\ x+2y+1=0
xy
C
2x+y+3=0 and x+2y+1=02x+y+3=0\ \text{and}\ x+2y+1=0
xy
D
2x+y−2=0 and x+2y−1=02x+y-2=0\ \text{and}\ x+2y-1=0
xy
122
Match each equation in List-I with the curve it represents in List-II. List-IList-IIP)x2+xy+y2−3x=01)a circleQ)xy−2x−3y+5=02)an ellipseR)xy−2x−3y+6=03)a parabolaS)x2+2xy+y2−4x+2y+3=04)a hyperbola5)a pair of straight lines\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^2+xy+y^2-3x=0 & \text{1)} \text{a circle} \\ \text{Q)} xy-2x-3y+5=0 & \text{2)} \text{an ellipse} \\ \text{R)} xy-2x-3y+6=0 & \text{3)} \text{a parabola} \\ \text{S)} x^2+2xy+y^2-4x+2y+3=0 & \text{4)} \text{a hyperbola} \\ & \text{5)} \text{a pair of straight lines} \end{array}
Not attempted
A
P-1, Q-4, R-5, S-3
B
P-2, Q-5, R-4, S-3
C
P-2, Q-4, R-5, S-3
D
P-2, Q-4, R-3, S-5
123
For each quadratic form in List-I, the xyxy-term can be removed by rotating the axes anticlockwise through a certain smallest positive angle. Match each form with that angle in List-II. List-IList-IIP)x2+4xy+y21)15∘Q)x2+23 xy−y22)22.5∘R)3x2+2xy+y23)30∘S)x2+23 xy+3y24)45∘5)60∘\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^2+4xy+y^2 & \text{1)} 15^\circ \\ \text{Q)} x^2+2\sqrt3\,xy-y^2 & \text{2)} 22.5^\circ \\ \text{R)} 3x^2+2xy+y^2 & \text{3)} 30^\circ \\ \text{S)} x^2+2\sqrt3\,xy+3y^2 & \text{4)} 45^\circ \\ & \text{5)} 60^\circ \end{array}
Not attempted
A
P-4, Q-3, R-5, S-2
B
P-3, Q-4, R-2, S-5
C
P-4, Q-2, R-3, S-5
D
P-4, Q-3, R-2, S-5
124
The axes are rotated anticlockwise through 30∘30^\circ about the origin. The curve x2+23 xy−y2=4x^2+2\sqrt3\,xy-y^2=4 then has the equation (in the new coordinates X,YX,Y)
Not attempted
A
X2−Y2=2X^2-Y^2=2
B
Y2−X2=2Y^2-X^2=2
C
X2−Y2=4X^2-Y^2=4
D
XY=2XY=2
125
A point PP moves so that the product of its perpendicular distances from the lines 3x−4y=03x-4y=0 and 4x+3y=04x+3y=0 is 66. The locus consists of two hyperbolas. The length of the transverse axis of each is
Not attempted
A
262\sqrt6
B
434\sqrt3
C
464\sqrt6
D
232\sqrt3
126
Consider the equation 4x2+4xy+y2−12x−6y+λ=04x^2+4xy+y^2-12x-6y+\lambda=0 where λ\lambda is a real constant. Which of the following statements are correct?
Not attempted
A
For λ=5\lambda=5 it represents two distinct parallel lines.
B
For λ=9\lambda=9 it represents a single straight line.
C
For λ=10\lambda=10 it has no real points.
D
For some real value of λ\lambda it represents a parabola.
127
Let S′(−5,0)S'(-5,0) be a focus of the hyperbola x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1. The number of points PP of the hyperbola (on both branches) with S′P=8S'P=8 is
Not attempted
xySS'
A
22
B
44
C
11
D
33
128
The length of the latus rectum of a hyperbola is equal to the distance between its foci. The eccentricity of the hyperbola is
Not attempted
A
1+52\frac{1+\sqrt5}{2}
B
2\sqrt2
C
22
D
5−12\frac{\sqrt5-1}{2}
129
Consider the hyperbola x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1. Which of the following statements are correct?
Not attempted
A
The length of its latus rectum is 92\frac92.
B
Its directrices are the lines x=±54x=\pm\frac{5}{4}.
C
The eccentricity of its conjugate hyperbola is 53\frac53.
D
Its asymptotes are y=±43xy=\pm\frac43x.
130
The distance between the foci of the hyperbola xy=8xy=8 is
Not attempted
A
88
B
828\sqrt2
C
424\sqrt2
D
1616
131
Let H: x216−y29=1H:\ \frac{x^2}{16}-\frac{y^2}{9}=1 and let H′H' be its conjugate hyperbola. Which of the following statements are correct?
Not attempted
A
HH and H′H' have the same pair of asymptotes.
B
If ee and e′e' are the eccentricities of HH and H′H', then 1e2+1e′2=1\frac1{e^2}+\frac1{e'^2}=1.
C
The four foci of HH and H′H' lie on the circle x2+y2=25x^2+y^2=25.
D
The length of the latus rectum of H′H' is 323\frac{32}{3}.
132
A point PP moves so that ∣PA−PB∣=6|PA-PB|=6, where A=(1,3)A=(1,3) and B=(1,−5)B=(1,-5). The locus of PP is the curve
Not attempted
A
(x−1)29−(y+1)27=1\frac{(x-1)^2}{9}-\frac{(y+1)^2}{7}=1
xyAB
B
(y+1)29−(x−1)216=1\frac{(y+1)^2}{9}-\frac{(x-1)^2}{16}=1
xyAB
C
(y+1)29−(x−1)27=1\frac{(y+1)^2}{9}-\frac{(x-1)^2}{7}=1
xyAB
D
(y−1)29−(x+1)27=1\frac{(y-1)^2}{9}-\frac{(x+1)^2}{7}=1
xyAB
133
A hyperbola has the lines x−2y+1=0x-2y+1=0 and x+2y−3=0x+2y-3=0 as its asymptotes and passes through the point (5,1)(5,1). Its eccentricity is
Not attempted
xyP
A
5\sqrt5
B
32\frac32
C
54\frac54
D
52\frac{\sqrt5}{2}
134
Match each hyperbola in List-I with its eccentricity in List-II. List-IList-IIP)x2−y2=41)54Q)9x2−16y2=1442)2R)y29−x216=13)53S)3x2−y2=34)25)32\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} x^2-y^2=4 & \text{1)} \frac54 \\ \text{Q)} 9x^2-16y^2=144 & \text{2)} \sqrt2 \\ \text{R)} \frac{y^2}{9}-\frac{x^2}{16}=1 & \text{3)} \frac53 \\ \text{S)} 3x^2-y^2=3 & \text{4)} 2 \\ & \text{5)} \frac32 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-2, Q-1, R-4, S-3
C
P-1, Q-2, R-3, S-4
D
P-2, Q-5, R-3, S-4
135
The circle having the line segment joining the foci of the hyperbola x24−y25=1\frac{x^2}{4}-\frac{y^2}{5}=1 as a diameter meets the hyperbola in four points. Find the area of the rectangle formed by these four points (correct to 2 decimal places).
Not attempted
xySS'
Your answer: —
Correct answer: 16.63
136
Let S(5,0)S(5,0) be a focus of the hyperbola x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1 and let A=(8,4)A=(8,4). A point PP moves on the right branch of the hyperbola. Find the minimum value of PA+PSPA+PS (correct to 2 decimal places).
Not attempted
xySS'A
Your answer: —
Correct answer: 7.6
137
Consider the rectangular hyperbola xy=4xy=4. Which of the following statements are correct?
Not attempted
A
The tangent at (4,1)(4,1) forms with the coordinate axes a triangle of area 1616.
B
The point (4,1)(4,1) is the midpoint of the part of its tangent lying between the axes.
C
The normal at (2,2)(2,2) is the line y=xy=x.
D
The length of the latus rectum is 44.
138
For the rectangular hyperbola xy=18xy=18, match each quantity in List-I with its value in List-II. List-IList-IIP)The length of the transverse axis1)12Q)The distance between the foci2)122R)The distance of the centre from a directrix3)32S)The distance of a focus from an asymptote4)65)62\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The length of the transverse axis} & \text{1)} 12 \\ \text{Q)} \text{The distance between the foci} & \text{2)} 12\sqrt2 \\ \text{R)} \text{The distance of the centre from a directrix} & \text{3)} 3\sqrt2 \\ \text{S)} \text{The distance of a focus from an asymptote} & \text{4)} 6 \\ & \text{5)} 6\sqrt2 \end{array}
Not attempted
A
P-1, Q-2, R-4, S-3
B
P-1, Q-2, R-3, S-4
C
P-2, Q-1, R-3, S-4
D
P-1, Q-5, R-3, S-4
139
A tangent is drawn to the hyperbola (x−1)(y−2)=6(x-1)(y-2)=6 at any point. The area of the triangle formed by this tangent and the two asymptotes is
Not attempted
xy
A
66
B
2424
C
1212
D
3636
140
Consider the hyperbola x29−y24=1\frac{x^2}{9}-\frac{y^2}{4}=1. Which of the following statements are correct?
Not attempted
A
A real tangent of slope mm exists only if ∣m∣>23|m|>\frac23.
B
The lines y=±23xy=\pm\frac23x are tangents to the hyperbola.
C
Exactly two tangents can be drawn from the point (0,1)(0,1).
D
Exactly two tangents can be drawn from the point (4,0)(4,0).
141
Which one of the following hyperbolas is such that two perpendicular tangents can be drawn to it from some point of the plane?
Not attempted
A
x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1
B
x2−y2=4x^2-y^2=4
C
y29−x216=1\frac{y^2}{9}-\frac{x^2}{16}=1
D
x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1
142
The normal at the point (5,163)\left(5,\frac{16}{3}\right) of the hyperbola x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1 meets the xx-axis at the point
Not attempted
A
(1259,0)\left(\frac{125}{9},0\right)
B
(95,0)\left(\frac{9}{5},0\right)
C
(259,0)\left(\frac{25}{9},0\right)
D
(−1259,0)\left(-\frac{125}{9},0\right)
143
Tangents are drawn to the hyperbola x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1 from a variable point on its directrix x=95x=\frac95. The chord of contact always passes through the fixed point
Not attempted
A
(95,0)\left(\frac95,0\right)
B
(5,0)(5,0)
C
(−5,0)(-5,0)
D
(259,0)\left(\frac{25}{9},0\right)
144
Let PP be a point on the hyperbola x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1. Which of the following statements are correct?
Not attempted
A
The product of the perpendicular distances of PP from the two asymptotes is 14425\frac{144}{25}.
B
The part of the tangent at PP lying between the asymptotes is bisected at PP.
C
The area of the triangle formed by the tangent at PP and the two asymptotes is 1212.
D
The area of the parallelogram formed by the asymptotes and the lines through PP parallel to them is 66.
145
Two tangents are drawn from the point P(5,1)P(5,1) to the hyperbola xy=9xy=9, touching it at AA and BB. The area of triangle PABPAB is
Not attempted
xyP
A
245\frac{24}{5}
B
965\frac{96}{5}
C
485\frac{48}{5}
D
99
146
For the hyperbola x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1, match each quantity in List-I with its value in List-II. List-IList-IIP)The positive value of c for which y=x+c is a tangent1)7Q)The product of the perpendicular distances of the foci from any tangent2)9R)The x-intercept of the tangent at the point (5,94)3)165S)The angle between the asymptotes (the one containing the transverse axis)4)2tan⁡−1345)25\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} \text{The positive value of } c \text{ for which } y=x+c \text{ is a tangent} & \text{1)} \sqrt7 \\ \text{Q)} \text{The product of the perpendicular distances of the foci from any tangent} & \text{2)} 9 \\ \text{R)} \text{The } x \text{-intercept of the tangent at the point } \left(5,\frac94\right) & \text{3)} \frac{16}{5} \\ \text{S)} \text{The angle between the asymptotes (the one containing the transverse axis)} & \text{4)} 2\tan^{-1}\frac34 \\ & \text{5)} 25 \end{array}
Not attempted
A
P-2, Q-1, R-3, S-4
B
P-1, Q-2, R-4, S-3
C
P-1, Q-3, R-2, S-4
D
P-1, Q-2, R-3, S-4
147
For the hyperbola x29−y216=1\frac{x^2}{9}-\frac{y^2}{16}=1, match the point of contact in List-I with the equation of the tangent at that point in List-II. List-IList-IIP)(3,0)1)x=3Q)(5,163)2)5x−3y=9R)(−5,163)3)5x+3y=−9S)(6,43)4)8x−33 y=125)5x+3y=9\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{P)} (3,0) & \text{1)} x=3 \\ \text{Q)} \left(5,\frac{16}{3}\right) & \text{2)} 5x-3y=9 \\ \text{R)} \left(-5,\frac{16}{3}\right) & \text{3)} 5x+3y=-9 \\ \text{S)} \left(6,4\sqrt3\right) & \text{4)} 8x-3\sqrt3\,y=12 \\ & \text{5)} 5x+3y=9 \end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-1, Q-3, R-2, S-4
C
P-1, Q-2, R-4, S-3
D
P-1, Q-5, R-3, S-4
148
For the hyperbola x29−y24=1\frac{x^2}{9}-\frac{y^2}{4}=1, consider the chord whose midpoint is (6,1)(6,1). Which of the following statements are correct?
Not attempted
A
The chord is the line 8x−3y=458x-3y=45.
B
The midpoints of all chords parallel to this chord lie on the line x=6yx=6y.
C
The tangents at the end points of this chord meet at the point (85,415)\left(\frac85,\frac4{15}\right).
D
The chord is parallel to neither of the asymptotes.
149
The foot of the perpendicular from the focus (5,0)(5,0) of the hyperbola x216−y29=1\frac{x^2}{16}-\frac{y^2}{9}=1 to its tangent y=x+7y=x+\sqrt7 is the point
Not attempted
A
(5+72,5−72)\left(\frac{5+\sqrt7}{2},\frac{5-\sqrt7}{2}\right)
B
(5−72,5+72)\left(\frac{5-\sqrt7}{2},\frac{5+\sqrt7}{2}\right)
C
(5−72,7−52)\left(\frac{5-\sqrt7}{2},\frac{\sqrt7-5}{2}\right)
D
(5−74,5+74)\left(\frac{5-\sqrt7}{4},\frac{5+\sqrt7}{4}\right)
150
Let PP be a point of the rectangular hyperbola x2−y2=8x^2-y^2=8 whose ordinate is 33, and let SS and S′S' be the foci. Then SP⋅S′PSP\cdot S'P equals
Not attempted
xySS'
A
1717
B
2525
C
2626
D
3434
For a point (x,y)(x,y) of a hyperbola, SP⋅S′P=e2x2−a2SP\cdot S'P=e^2x^2-a^2. What is special when e2=2e^2=2?
For x2−y2=8x^2-y^2=8: a2=b2=8a^2=b^2=8, e2=2e^2=2. For P(x,y)P(x,y) on the right branch, SP=ex−aSP=ex-a and S′P=ex+aS'P=ex+a, so SP⋅S′P=e2x2−a2=2x2−8SP\cdot S'P=e^2x^2-a^2=2x^2-8. With y=3y=3: x2=8+9=17x^2=8+9=17, so SP⋅S′P=34−8=26SP\cdot S'P=34-8=26. This equals OP2=x2+y2=17+9=26OP^2=x^2+y^2=17+9=26: for a rectangular hyperbola SP⋅S′P=OP2SP\cdot S'P=OP^2 (since 2x2−8=x2+y22x^2-8=x^2+y^2 when x2−y2=8x^2-y^2=8). The value x2=17x^2=17 is OP2OP^2 without the y2y^2 term.