Clear Notion
Login

Solve PYQs

JEE Advanced · Mathematics · Three Dimensional Geometry

Your Score

0 / 112

Correct

0 / 28

Attempted

0 / 28

1
Let LL be the line through P(1,2,−1)P(1,2,-1) and Q(2,3,1)Q(2,3,1). Let SS be the foot of the perpendicular from R(4,−1,5)R(4,-1,5) to LL. Another line through RR meets LL at TT such that SS divides PTPT internally in the ratio ∣PS∣:∣ST∣=1:2|PS|:|ST|=1:2. Then which of the following statements is (are) TRUE?
Not attempted
A
The orthocentre of triangle PRTPRT is (235,−4,315)\left(\dfrac{23}5,-4,\dfrac{31}5\right)
B
The orthocentre of triangle PRTPRT is (4,3,5)(4,3,5)
C
The area of triangle PRTPRT is 656\sqrt5
D
The area of triangle PRTPRT is 18518\sqrt5
2
Let PP be the plane containing the line x−12=y−33=z+21\dfrac{x-1}{2}=\dfrac{y-3}{3}=\dfrac{z+2}{1} and perpendicular to the plane x+2y+3z=4x+2y+3z=4. Let P1P_1 be the plane through (4,2,2)(4,2,2) parallel to PP. Then which of the following statements is (are) TRUE?
Not attempted
A
The equation of the plane PP is 7x−5y+z=−107x-5y+z=-10
B
The distance between the planes PP and P1P_1 is 3030
C
The distance of the plane PP from the origin is 232\sqrt3
D
The acute angle between PP and 2x+2y+z=32x+2y+z=3 is cos⁡−1 ⁣(133)\cos^{-1}\!\left(\dfrac{1}{3\sqrt3}\right)
3
Let L1L_1 be the line of intersection of the planes given by the equations 2x+3y+z=4andx+2y+z=52x+3y+z=4 \quad \text{and} \quad x+2y+z=5. Let L2L_2 be the line passing through the point P(2,−1,3)P(2,-1,3) and parallel to L1L_1. Let MM denote the plane given by the equation 2x+y−2z=62x+y-2z=6. Suppose that the line L2L_2 meets the plane MM at the point QQ. Let RR be the foot of the perpendicular drawn from PP to the plane MM. Then which of the following statements is (are) TRUE?
Not attempted
A
The length of the line segment PQPQ is 939\sqrt{3}
B
The length of the line segment QRQR is 1515
C
The area of △PQR\triangle PQR is 32234\dfrac{3}{2}\sqrt{234}
D
The acute angle between the line segments PQPQ and PRPR is cos⁡−1 ⁣(123)\cos^{-1}\!\left(\dfrac{1}{2\sqrt{3}}\right)
4
A straight line drawn from the point P(1,3,2)P(1,3,2), parallel to the line x−21=y−42=z−61\dfrac{x-2}{1}=\dfrac{y-4}{2}=\dfrac{z-6}{1}, intersects the plane L1:x−y+3z=6L_1: x-y+3z=6 at the point QQ. Another straight line which passes through QQ and is perpendicular to the plane L1L_1 intersects the plane L2:2x−y+z=−4L_2: 2x-y+z=-4 at the point RR. Then which of the following statements is (are) TRUE?
Not attempted
A
The length of the line segment PQPQ is 6\sqrt6
B
The coordinates of RR are (1,6,3)(1,6,3)
C
The centroid of the triangle PQRPQR is (43,143,53)\left(\dfrac43,\dfrac{14}3,\dfrac53\right)
D
The perimeter of the triangle PQRPQR is 2+6+11\sqrt2+\sqrt6+\sqrt{11}
5
Let R3\mathbb R^3 denote the three-dimensional space. Take two points P=(1,2,3)P=(1,2,3) and Q=(4,2,7)Q=(4,2,7). Let dist(X,Y)\text{dist}(X,Y) denote the distance between two points XX and YY in R3\mathbb R^3. Let S={X∈R3:(dist(X,P))2−(dist(X,Q))2=50}S=\{X\in\mathbb R^3 : (\text{dist}(X,P))^2-(\text{dist}(X,Q))^2=50\} and T={Y∈R3:(dist(Y,Q))2−(dist(Y,P))2=50}T=\{Y\in\mathbb R^3 : (\text{dist}(Y,Q))^2-(\text{dist}(Y,P))^2=50\}. Then which of the following statements is (are) TRUE?
Not attempted
A
There is a triangle whose area is 11 and all of whose vertices are from SS.
B
There are two distinct points LL and MM in TT such that each point on segment LMLM is also in TT.
C
There are infinitely many rectangles of perimeter 4848, two of whose vertices are from SS and the other two from TT.
D
There is a square of perimeter 4848, two of whose vertices are from SS and the other two from TT.
6
Let γ∈R\gamma\in\mathbb R be such that the lines L1:x+111=y+212=z+293L_1:\dfrac{x+11}{1}=\dfrac{y+21}{2}=\dfrac{z+29}{3} and L2:x+163=y+112=z+4γL_2:\dfrac{x+16}{3}=\dfrac{y+11}{2}=\dfrac{z+4}{\gamma} intersect. Let R1R_1 be the point of intersection of L1L_1 and L2L_2. Let O=(0,0,0)O=(0,0,0), and n^\hat n denote a unit normal vector to the plane containing both lines L1L_1 and L2L_2. Match each entry in List-I to the correct entry in List-II. List-I: (P) γ\gamma equals (Q) A possible choice for n^\hat n is (R) OR1→\overrightarrow{OR_1} equals (S) A possible value of OR1→⋅n^\overrightarrow{OR_1}\cdot\hat n is List-II: (1) −i^−j^+k^-\hat i-\hat j+\hat k (2) 3/2\sqrt{3/2} (3) 11 (4) 16i^−26j^+16k^\dfrac1{\sqrt6}\hat i-\dfrac2{\sqrt6}\hat j+\dfrac1{\sqrt6}\hat k (5) 2/3\sqrt{2/3} The correct option is
Not attempted
A
(P)→\to(3) (Q)→\to(4) (R)→\to(1) (S)→\to(2)
B
(P)→\to(5) (Q)→\to(4) (R)→\to(1) (S)→\to(2)
C
(P)→\to(3) (Q)→\to(4) (R)→\to(1) (S)→\to(5)
D
(P)→\to(3) (Q)→\to(1) (R)→\to(4) (S)→\to(5)
7
Let ℓ1\ell_1 and ℓ2\ell_2 be the lines r⃗1=λ(i^+j^+k^)\vec r_1=\lambda(\hat i+\hat j+\hat k) and r⃗2=(j^−k^)+μ(i^+k^)\vec r_2=(\hat j-\hat k)+\mu(\hat i+\hat k), respectively. Let XX be the set of all the planes HH that contain the line ℓ1\ell_1. For a plane HH, let d(H)d(H) denote the smallest possible distance between the points of ℓ2\ell_2 and HH. Let H0H_0 be a plane in XX for which d(H0)d(H_0) is the maximum value of d(H)d(H) as HH varies over all planes in XX. Match each entry in List-I to the correct entries in List-II. List-I (P) The value of d(H0)d(H_0) is (Q) The distance of the point (0,1,2)(0,1,2) from H0H_0 is (R) The distance of origin from H0H_0 is (S) The distance of origin from the point of intersection of planes y=zy=z, x=1x=1 and H0H_0 is List-II (1) 3\sqrt3 (2) 13\dfrac1{\sqrt3} (3) 00 (4) 2\sqrt2 (5) 12\dfrac1{\sqrt2} The correct option is:
Not attempted
A
(P)→(2)(Q)→(4)(R)→(5)(S)→(1)(P)\to(2)\quad(Q)\to(4)\quad(R)\to(5)\quad(S)\to(1)
B
(P)→(5)(Q)→(4)(R)→(3)(S)→(1)(P)\to(5)\quad(Q)\to(4)\quad(R)\to(3)\quad(S)\to(1)
C
(P)→(2)(Q)→(1)(R)→(3)(S)→(2)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(3)\quad(S)\to(2)
D
(P)→(5)(Q)→(1)(R)→(4)(S)→(2)(P)\to(5)\quad(Q)\to(1)\quad(R)\to(4)\quad(S)\to(2)
8
Let QQ be the cube with the set of vertices {(x1,x2,x3)∈R3:x1,x2,x3∈{0,1}}\{(x_1,x_2,x_3)\in\mathbb{R}^3 : x_1,x_2,x_3\in\{0,1\}\}. Let FF be the set of all twelve lines containing the diagonals of the six faces of the cube QQ. Let SS be the set of all four lines containing the main diagonals of the cube QQ; for instance, the line passing through the vertices (0,0,0)(0,0,0) and (1,1,1)(1,1,1) is in SS. For lines ℓ1\ell_1 and ℓ2\ell_2, let d(ℓ1,ℓ2)d(\ell_1,\ell_2) denote the shortest distance between them. Then the maximum value of d(ℓ1,ℓ2)d(\ell_1,\ell_2), as ℓ1\ell_1 varies over FF and ℓ2\ell_2 varies over SS, is
Not attempted
A
16\dfrac{1}{\sqrt6}
B
18\dfrac{1}{\sqrt8}
C
13\dfrac{1}{\sqrt3}
D
112\dfrac{1}{\sqrt{12}}
9
Let P1P_1 and P2P_2 be two planes given by P1:10x+15y+12z−60=0P_1: 10x+15y+12z-60=0, P2:−2x+5y+4z−20=0P_2: -2x+5y+4z-20=0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1P_1 and P2P_2?
Not attempted
A
x−10=y−10=z−15\dfrac{x-1}{0}=\dfrac{y-1}{0}=\dfrac{z-1}{5}
B
x−6−5=y2=z3\dfrac{x-6}{-5}=\dfrac{y}{2}=\dfrac{z}{3}
C
x−2=y−45=z4\dfrac{x}{-2}=\dfrac{y-4}{5}=\dfrac{z}{4}
D
x1=y−4−2=z3\dfrac{x}{1}=\dfrac{y-4}{-2}=\dfrac{z}{3}
10
Let SS be the reflection of a point QQ with respect to the plane given by r⃗=−(t+p)i^+tj^+(1+p)k^\vec r = -(t+p)\hat i+t\hat j+(1+p)\hat k where t,pt,p are real parameters and i^,j^,k^\hat i,\hat j,\hat k are the unit vectors along the three positive coordinate axes. If the position vectors of QQ and SS are 10i^+15j^+20k^10\hat i+15\hat j+20\hat k and αi^+βj^+γk^\alpha\hat i+\beta\hat j+\gamma\hat k respectively, then which of the following is/are TRUE?
Not attempted
A
3(α+β)=−1013(\alpha+\beta)=-101
B
3(β+γ)=−713(\beta+\gamma)=-71
C
3(γ+α)=−863(\gamma+\alpha)=-86
D
3(α+β+γ)=−1213(\alpha+\beta+\gamma)=-121
11
Let α,β\alpha, \beta and γ\gamma be real numbers such that the system of linear equations x+2y+3z=αx+2y+3z=\alpha 4x+5y+6z=β4x+5y+6z=\beta 7x+8y+9z=γ−17x+8y+9z=\gamma-1 is consistent. Let ∣M∣|M| represent the determinant of the matrix M=(α2γβ10−101)M=\begin{pmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{pmatrix} Let PP be the plane containing all those (α,β,γ)(\alpha,\beta,\gamma) for which the above system of linear equations is consistent, and DD be the square of the distance of the point (0,1,0)(0,1,0) from the plane PP. The value of DD is ____.
Not attempted
Your answer: —
Correct answer: 1.5
12
Let α,β,γ,δ\alpha,\beta,\gamma,\delta be real numbers such that α2+β2+γ2≠0\alpha^2+\beta^2+\gamma^2 \ne 0 and α+γ=1\alpha+\gamma=1. Suppose the point (3,2,−1)(3,2,-1) is the mirror image of the point (1,0,−1)(1,0,-1) with respect to the plane αx+βy+γz=δ\alpha x+\beta y+\gamma z=\delta. Then which of the following statements is/are TRUE?
Not attempted
A
α+β=2\alpha+\beta=2
B
δ−γ=3\delta-\gamma=3
C
δ+β=4\delta+\beta=4
D
α+β+γ=δ\alpha+\beta+\gamma=\delta
13
Let L1L_1 and L2L_2 be the following straight lines. L1:x−11=y−1=z−13L_1: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-1}{3} and L2:x−1−3=y−1=z−11L_2: \frac{x-1}{-3} = \frac{y}{-1} = \frac{z-1}{1}. Suppose the straight line L:x−αl=y−1m=z−γ−2L: \frac{x-\alpha}{l} = \frac{y-1}{m} = \frac{z-\gamma}{-2} lies in the plane containing L1L_1 and L2L_2, and passes through the point of intersection of L1L_1 and L2L_2. If the line LL bisects the acute angle between the lines L1L_1 and L2L_2, then which of the following statements is/are TRUE?
Not attempted
A
α−γ=3\alpha - \gamma = 3
B
l+m=2l+m = 2
C
α−γ=1\alpha - \gamma = 1
D
l+m=0l+m = 0
14
Three lines L1:r⃗=λi^, λ∈RL_1: \vec r = \lambda\hat i,\ \lambda\in\mathbb{R}, L2:r⃗=k^+μj^, μ∈RL_2: \vec r = \hat k+\mu\hat j,\ \mu\in\mathbb{R}, and L3:r⃗=i^+j^+νk^, ν∈RL_3: \vec r = \hat i+\hat j+\nu\hat k,\ \nu\in\mathbb{R} are given. For which point(s) QQ on L2L_2 can we find a point PP on L1L_1 and a point RR on L3L_3 so that PP, QQ and RR are collinear?
Not attempted
A
k^+j^\hat k+\hat j
B
k^\hat k
C
k^+12j^\hat k+\tfrac12\hat j
D
k^−12j^\hat k-\tfrac12\hat j
15
Let L1L_1 and L2L_2 denote the lines r⃗=i^+λ(−i^+2j^+2k^), λ∈R\vec r=\hat i+\lambda(-\hat i+2\hat j+2\hat k),\ \lambda\in\mathbb{R} and r⃗=μ(2i^−j^+2k^), μ∈R\vec r=\mu(2\hat i-\hat j+2\hat k),\ \mu\in\mathbb{R} respectively. If L3L_3 is a line which is perpendicular to both L1L_1 and L2L_2 and cuts both of them, then which of the following options describe(s) L3L_3?
Not attempted
A
r⃗=13(2i^+k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac13(2\hat i+\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
B
r⃗=29(2i^−j^+2k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac29(2\hat i-\hat j+2\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
C
r⃗=t(2i^+2j^−k^), t∈R\vec r=t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
D
r⃗=29(4i^+j^+k^)+t(2i^+2j^−k^), t∈R\vec r=\dfrac29(4\hat i+\hat j+\hat k)+t(2\hat i+2\hat j-\hat k),\ t\in\mathbb{R}
16
Three lines are given by r⃗=λi^, λ∈R\vec r=\lambda\hat i,\ \lambda\in\mathbb{R}, r⃗=μ(i^+j^), μ∈R\vec r=\mu(\hat i+\hat j),\ \mu\in\mathbb{R} and r⃗=ν(i^+j^+k^), ν∈R\vec r=\nu(\hat i+\hat j+\hat k),\ \nu\in\mathbb{R}. Let the lines cut the plane x+y+z=1x+y+z=1 at the points AA, BB and CC respectively. If the area of the triangle ABCABC is Δ\Delta then the value of (6Δ)2(6\Delta)^2 equals _____
Not attempted
Your answer: —
Correct answer: 0.75
17
Let PP be a point in the first octant, whose image QQ in the plane x+y=3x+y=3 (that is, the line segment PQPQ is perpendicular to the plane x+y=3x+y=3 and the mid-point of PQPQ lies in the plane x+y=3x+y=3) lies on the zz-axis. Let the distance of PP from the xx-axis be 55. If RR is the image of PP in the xyxy-plane, then the length of RPRP is _____.
Not attempted
Your answer: —
Correct answer: 8
18
Let P1:2x+y−z=3P_1: 2x+y-z=3 and P2:x+2y+z=2P_2: x+2y+z=2 be two planes. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
The line of intersection of P1P_1 and P2P_2 has direction ratios 1,2,−11,2,-1
B
The line 3x−49=1−3y9=z3\dfrac{3x-4}{9}=\dfrac{1-3y}{9}=\dfrac{z}{3} is perpendicular to the line of intersection of P1P_1 and P2P_2
C
The acute angle between P1P_1 and P2P_2 is 60∘60^\circ
D
If P3P_3 is the plane passing through the point (4,2,−2)(4,2,-2) and perpendicular to the line of intersection of P1P_1 and P2P_2, then the distance of the point (2,1,1)(2,1,1) from the plane P3P_3 is 23\dfrac{2}{\sqrt3}
19
The equation of the plane passing through the point (1,1,1)(1, 1, 1) and perpendicular to the planes 2x+y−2z=52x + y - 2z = 5 and 3x−6y−2z=73x - 6y - 2z = 7, is
Not attempted
A
14x+2y−15z=114x + 2y - 15z = 1
B
14x−2y+15z=2714x - 2y + 15z = 27
C
14x+2y+15z=3114x + 2y + 15z = 31
D
−14x+2y+15z=3-14x + 2y + 15z = 3
20
Let PP be the image of the point (3,1,7)(3,1,7) with respect to the plane x−y+z=3x-y+z=3. Then the equation of the plane passing through PP and containing the straight line x1=y2=z1\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{1} is
Not attempted
A
x+y−3z=0x+y-3z=0
B
3x+z=03x+z=0
C
x−4y+7z=0x-4y+7z=0
D
2x−y=02x-y=0
21
Consider a pyramid OPQRSOPQRS located in the first octant (x≥0,y≥0,z≥0x\ge0,y\ge0,z\ge0) with OO as origin, and OPOP and OROR along the xx-axis and the yy-axis, respectively. The base OPQROPQR of the pyramid is a square with OP=3OP=3. The point SS is directly above the mid-point TT of diagonal OQOQ such that TS=3TS=3. Then
Not attempted
A
the acute angle between OQOQ and OSOS is π3\dfrac{\pi}{3}
B
the equation of the plane containing the triangle OQSOQS is x−y=0x-y=0
C
the length of the perpendicular from PP to the plane containing the triangle OQSOQS is 32\dfrac{3}{\sqrt2}
D
the perpendicular distance from OO to the straight line containing RSRS is 152\sqrt{\dfrac{15}{2}}
22
In R3\mathbb{R}^3, consider the planes P1:y=0P_1: y=0 and P2:x+z=1P_2: x+z=1. Let P3P_3 be a plane, different from P1P_1 and P2P_2, which passes through the intersection of P1P_1 and P2P_2. If the distance of the point (0,1,0)(0,1,0) from P3P_3 is 11 and the distance of a point (α,β,γ)(\alpha,\beta,\gamma) from P3P_3 is 22, then which of the following relations is (are) true?
Not attempted
A
2α+β+2γ+2=02\alpha+\beta+2\gamma+2=0
B
2α−β+2γ+4=02\alpha-\beta+2\gamma+4=0
C
2α+β−2γ−10=02\alpha+\beta-2\gamma-10=0
D
2α−β+2γ−8=02\alpha-\beta+2\gamma-8=0
23
In R3\mathbb{R}^3, let LL be a straight line passing through the origin. Suppose that all the points on LL are at a constant distance from the two planes P1:x+2y−z+1=0P_1: x+2y-z+1=0 and P2:2x−y+z−1=0P_2: 2x-y+z-1=0. Let MM be the locus of the feet of the perpendiculars drawn from the points on LL to the plane P1P_1. Which of the following points lie(s) on MM?
Not attempted
A
(0,−56,−23)\left(0,-\dfrac56,-\dfrac23\right)
B
(−16,−13,16)\left(-\dfrac16,-\dfrac13,\dfrac16\right)
C
(−56,0,16)\left(-\dfrac56,0,\dfrac16\right)
D
(−13,0,23)\left(-\dfrac13,0,\dfrac23\right)
24
From a point P(λ,λ,λ)P(\lambda,\lambda,\lambda), perpendiculars PQPQ and PRPR are drawn respectively on the lines y=x, z=1y=x,\,z=1 and y=−x, z=−1y=-x,\,z=-1. If PP is such that ∠QPR\angle QPR is a right angle, then the possible value(s) of λ\lambda is(are):
Not attempted
A
2\sqrt{2}
B
11
C
−1-1
D
−2-\sqrt{2}
25
Two lines L1:x=5, y3−α=z−2L_1: x=5,\ \dfrac{y}{3-\alpha}=\dfrac{z}{-2} and L2:x=α, y−1=z2−αL_2: x=\alpha,\ \dfrac{y}{-1}=\dfrac{z}{2-\alpha} are coplanar. Then α\alpha can take value(s)
Not attempted
A
11
B
22
C
33
D
44
26
Consider the lines L1:x−12=y−1=z+31L_1: \dfrac{x-1}{2}=\dfrac{y}{-1}=\dfrac{z+3}{1}, L2:x−41=y+31=z+32L_2: \dfrac{x-4}{1}=\dfrac{y+3}{1}=\dfrac{z+3}{2} and the planes P1:7x+y+2z=3P_1: 7x+y+2z=3, P2:3x+5y−6z=4P_2: 3x+5y-6z=4. Let ax+by+cz=dax+by+cz=d be the equation of the plane passing through the point of intersection of lines L1L_1 and L2L_2, and perpendicular to planes P1P_1 and P2P_2. Match List I with List II and select the correct answer using the code given below the lists: List I: P. a=a= Q. b=b= R. c=c= S. d=d= List II: 1. 1313 2. −3-3 3. 11 4. −2-2 Codes:
Not attempted
A
(A) P→3, Q→2, R→4, S→1P\to3,\ Q\to2,\ R\to4,\ S\to1
B
(B) P→1, Q→3, R→4, S→2P\to1,\ Q\to3,\ R\to4,\ S\to2
C
(C) P→3, Q→2, R→1, S→4P\to3,\ Q\to2,\ R\to1,\ S\to4
D
(D) P→2, Q→4, R→1, S→3P\to2,\ Q\to4,\ R\to1,\ S\to3
27
A line ll passing through the origin is perpendicular to the lines l1:(3+t)i^+(−1+2t)j^+(4+2t)k^, −∞<t<∞l_1:(3+t)\hat i+(-1+2t)\hat j+(4+2t)\hat k,\ -\infty<t<\infty and l2:(3+2s)i^+(3+2s)j^+(2+s)k^, −∞<s<∞l_2:(3+2s)\hat i+(3+2s)\hat j+(2+s)\hat k,\ -\infty<s<\infty. Then, the coordinate(s) of the point(s) on l2l_2 at a distance of 17\sqrt{17} from the point of intersection of ll and l1l_1 is (are)
Not attempted
A
(73,73,53)\left(\dfrac73,\dfrac73,\dfrac53\right)
B
(−1,−1,0)(-1,-1,0)
C
(1,1,1)(1,1,1)
D
(79,79,89)\left(\dfrac79,\dfrac79,\dfrac89\right)
28
Perpendiculars are drawn from points on the line x+22=y+1−1=z3\dfrac{x+2}{2}=\dfrac{y+1}{-1}=\dfrac{z}{3} to the plane x+y+z=3x+y+z=3. The feet of perpendiculars lie on the line
Not attempted
A
x5=y−18=z−2−13\dfrac{x}{5}=\dfrac{y-1}{8}=\dfrac{z-2}{-13}
B
x2=y−13=z−2−5\dfrac{x}{2}=\dfrac{y-1}{3}=\dfrac{z-2}{-5}
C
x4=y−13=z−2−7\dfrac{x}{4}=\dfrac{y-1}{3}=\dfrac{z-2}{-7}
D
x2=y−1−7=z−25\dfrac{x}{2}=\dfrac{y-1}{-7}=\dfrac{z-2}{5}
Parametrize a general point on the given line, drop the foot of its perpendicular onto the plane using the standard formula foot=P−(P⋅n−d)∣n∣2n\text{foot}=P-\dfrac{(P\cdot n - d)}{|n|^2}n, then eliminate the parameter to get the locus.
Deconstruction\textbf{Deconstruction}: The feet of perpendiculars from every point of a line onto a plane themselves trace out a line; find its symmetric equation. Strategy\textbf{Strategy}: Take a general point on the given line with parameter tt, compute its perpendicular foot on the plane x+y+z=3x+y+z=3 (normal (1,1,1)(1,1,1)), then eliminate tt between the resulting coordinates. Execution\textbf{Execution}: General point on the line: P=(−2+2t, −1−t, 3t)P=(-2+2t,\,-1-t,\,3t). - Foot of perpendicular onto x+y+z=3x+y+z=3: Foot=P−(xP+yP+zP−3)3(1,1,1)\text{Foot}=P-\dfrac{(x_P+y_P+z_P-3)}{3}(1,1,1). - xP+yP+zP=(−2+2t)+(−1−t)+3t=−3+4tx_P+y_P+z_P=(-2+2t)+(-1-t)+3t=-3+4t, so the correction term uses (−3+4t−3)=−6+4t(-3+4t-3)=-6+4t. - X=−2+2t+6−4t3=2t3X=-2+2t+\dfrac{6-4t}{3}=\dfrac{2t}{3} - Y=−1−t+6−4t3=3−7t3=1−7t3Y=-1-t+\dfrac{6-4t}{3}=\dfrac{3-7t}{3}=1-\dfrac{7t}{3}, so Y−1=−7t3Y-1=-\dfrac{7t}{3} - Z=3t+6−4t3=6+5t3=2+5t3Z=3t+\dfrac{6-4t}{3}=\dfrac{6+5t}{3}=2+\dfrac{5t}{3}, so Z−2=5t3Z-2=\dfrac{5t}{3} - So X:(Y−1):(Z−2)=2t3:(−7t3):5t3=2:(−7):5X:(Y-1):(Z-2) = \dfrac{2t}{3}:\left(-\dfrac{7t}{3}\right):\dfrac{5t}{3} = 2:(-7):5. - Locus: x2=y−1−7=z−25\dfrac{x}{2}=\dfrac{y-1}{-7}=\dfrac{z-2}{5}. Verification\textbf{Verification}: At t=0t=0 the foot is (0,1,2)(0,1,2), which indeed satisfies x+y+z=0+1+2=3x+y+z=0+1+2=3 (lies on the plane) and also satisfies 02=1−1−7=2−25=0\dfrac{0}{2}=\dfrac{1-1}{-7}=\dfrac{2-2}{5}=0 (lies on the derived line) — a consistent check point. Final Answer\textbf{Final Answer}: (D) x2=y−1−7=z−25\dfrac{x}{2}=\dfrac{y-1}{-7}=\dfrac{z-2}{5}