Clear Notion
Login

Solve PYQs

JEE Advanced · Mathematics · Complex Numbers and Quadratic Equations

Your Score

0 / 140

Correct

0 / 35

Attempted

0 / 35

1
Let a,b,ca,b,c be positive integers in arithmetic progression such that ax2+bx+c=0ax^2+bx+c=0 has only integer solutions. Then which of the following statements is (are) TRUE?
Not attempted
A
c−bc-b is an integer multiple of aa
B
Both roots of ax2+bx+c=0ax^2+bx+c=0 are odd integers
C
If c=15c=15, then ab=8ab=8
D
If b=8b=8, then x=3x=3 is a root of ax2+bx+c=0ax^2+bx+c=0
2
Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I\textbf{List-I} (P) If α,β\alpha,\beta are the distinct roots of x2+x+1=0x^2+x+1=0, the quadratic equation with roots 1(α+1)2026\dfrac{1}{(\alpha+1)^{2026}} and 1(β+1)2026\dfrac{1}{(\beta+1)^{2026}} is (Q) If α,β\alpha,\beta are as above, the quadratic equation with roots 1(α+1)2027\dfrac{1}{(\alpha+1)^{2027}} and 1(β+1)2027\dfrac{1}{(\beta+1)^{2027}} is (R) If γ,δ\gamma,\delta are the distinct roots of x2−x+1=0x^2-x+1=0, the value of 1(γ−1)2026+1(δ−1)2026\dfrac{1}{(\gamma-1)^{2026}}+\dfrac{1}{(\delta-1)^{2026}} is (S) If p,rp,r are the distinct roots of x2+x−1=0x^2+x-1=0, the value of 1(p+1)3+1(r+1)3\dfrac{1}{(p+1)^3}+\dfrac{1}{(r+1)^3} is List-II\textbf{List-II} (1) x2+x+1=0x^2+x+1=0 (2) x2−x+1=0x^2-x+1=0 (3) x2+x−1=0x^2+x-1=0 (4) −1-1 (5) −4-4
Not attempted
A
P→1,Q→2,R→5,S→4P\to1,Q\to2,R\to5,S\to4
B
P→3,Q→1,R→4,S→5P\to3,Q\to1,R\to4,S\to5
C
P→1,Q→2,R→4,S→5P\to1,Q\to2,R\to4,S\to5
D
P→2,Q→3,R→5,S→4P\to2,Q\to3,R\to5,S\to4
3
For a non-zero complex number zz, let arg⁡(z)\arg(z) denote the principal argument of zz, with −π<arg⁡(z)≤π-\pi<\arg(z)\le\pi. Let ω\omega be the cube root of unity for which 0<arg⁡(ω)<π0<\arg(\omega)<\pi. Let α=arg⁡(∑n=12025(−ω)n).\alpha = \arg\left(\displaystyle\sum_{n=1}^{2025}(-\omega)^n\right). Then the value of 3απ\dfrac{3\alpha}{\pi} is ________________.
Not attempted
Your answer: —
Correct answer: -2
4
Let R\mathbb{R} denote the set of all real numbers. Let z1=1+2iz_1 = 1+2i and z2=3iz_2=3i be two complex numbers, where i=−1i=\sqrt{-1}. Let S={(x,y)∈R×R:∣x+iy−z1∣=2∣x+iy−z2∣}S = \{(x,y)\in\mathbb{R}\times\mathbb{R} : |x+iy-z_1| = 2|x+iy-z_2|\}. Then which of the following statements is (are) TRUE?
Not attempted
A
SS is a circle with centre (−13,103)\left(-\dfrac13,\dfrac{10}{3}\right)
B
SS is a circle with centre (13,83)\left(\dfrac13,\dfrac{8}{3}\right)
C
SS is a circle with radius 23\dfrac{\sqrt2}{3}
D
SS is a circle with radius 223\dfrac{2\sqrt2}{3}
5
Let f(x)=x4+ax3+bx2+cf(x)=x^4+ax^3+bx^2+c be a polynomial with real coefficients such that f(1)=−9f(1)=-9. Suppose that i3i\sqrt3 is a root of the equation 4x3+3ax2+2bx=04x^3+3ax^2+2bx=0, where i=−1i=\sqrt{-1}. If α1,α2,α3,α4\alpha_1,\alpha_2,\alpha_3,\alpha_4 are all the roots of the equation f(x)=0f(x)=0, then ∣α1∣2+∣α2∣2+∣α3∣2+∣α4∣2|\alpha_1|^2+|\alpha_2|^2+|\alpha_3|^2+|\alpha_4|^2 is equal to ______.
Not attempted
Your answer: —
Correct answer: 20
6
Let A1,A2,A3,…,A8A_1,A_2,A_3,\ldots,A_8 be the vertices of a regular octagon that lie on a circle of radius 22. Let PP be a point on the circle and let PAiPA_i denote the distance between PP and AiA_i for i=1,2,…,8i=1,2,\ldots,8. If PP varies over the circle, then the maximum value of the product PA1⋅PA2⋯PA8PA_1\cdot PA_2\cdots PA_8 is
Not attempted
Your answer: —
Correct answer: 512
7
Let zz be a complex number satisfying ∣z∣3+2z2+4zˉ−8=0|z|^3+2z^2+4\bar z-8=0, where zˉ\bar z denotes the complex conjugate of zz. Let the imaginary part of zz be nonzero. Match each entry in List-I to the correct entries in List-II. List-I (P) ∣z∣2|z|^2 is equal to (Q) ∣z−zˉ∣2|z-\bar z|^2 is equal to (R) ∣z∣2+∣z+zˉ∣2|z|^2+|z+\bar z|^2 is equal to (S) ∣z+1∣2|z+1|^2 is equal to List-II (1) 1212 (2) 44 (3) 88 (4) 1010 (5) 77 The correct option is:
Not attempted
A
(P)→(1)(Q)→(3)(R)→(5)(S)→(4)(P)\to(1)\quad(Q)\to(3)\quad(R)\to(5)\quad(S)\to(4)
B
(P)→(2)(Q)→(1)(R)→(3)(S)→(5)(P)\to(2)\quad(Q)\to(1)\quad(R)\to(3)\quad(S)\to(5)
C
(P)→(2)(Q)→(4)(R)→(5)(S)→(1)(P)\to(2)\quad(Q)\to(4)\quad(R)\to(5)\quad(S)\to(1)
D
(P)→(2)(Q)→(3)(R)→(5)(S)→(4)(P)\to(2)\quad(Q)\to(3)\quad(R)\to(5)\quad(S)\to(4)
8
Let A={1967+1686 isin⁡θ7−3icos⁡θ:θ∈R}A=\left\{\dfrac{1967+1686\,i\sin\theta}{7-3i\cos\theta} : \theta\in\mathbb{R}\right\}. If AA contains exactly one positive integer nn, then the value of nn is
Not attempted
Your answer: —
Correct answer: 281
9
Let zˉ\bar z denote the complex conjugate of a complex number zz. If zz is a non-zero complex number for which both real and imaginary parts of (zˉ)2+1z2(\bar z)^2+\dfrac{1}{z^2} are integers, then which of the following is/are possible value(s) of ∣z∣|z|?
Not attempted
A
(43+32052)1/4\left(\dfrac{43+3\sqrt{205}}{2}\right)^{1/4}
B
(7+334)1/4\left(\dfrac{7+\sqrt{33}}{4}\right)^{1/4}
C
(9+654)1/4\left(\dfrac{9+\sqrt{65}}{4}\right)^{1/4}
D
(7+136)1/4\left(\dfrac{7+\sqrt{13}}{6}\right)^{1/4}
10
The product of all positive real values of xx satisfying the equation x(16(log⁡5x)3−68log⁡5x)=5−16x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16} is _______.
Not attempted
Your answer: —
Correct answer: 1
11
Let zz be a complex number with non-zero imaginary part. If 2+3z+4z22−3z+4z2\dfrac{2+3z+4z^2}{2-3z+4z^2} is a real number, then the value of ∣z∣2|z|^2 is __________.
Not attempted
Your answer: —
Correct answer: 0.5
12
Let zˉ\bar z denote the complex conjugate of a complex number zz and let i=−1i=\sqrt{-1}. In the set of complex numbers, the number of distinct roots of the equation zˉ−z2=i(zˉ+z2)\bar z - z^2 = i(\bar z+z^2) is __________.
Not attempted
Your answer: —
Correct answer: 4
13
For any complex number w=c+idw=c+id, let arg⁡(w)∈(−π,π]\arg(w)\in(-\pi,\pi], where i=−1i=\sqrt{-1}. Let α\alpha and β\beta be real numbers such that for all complex numbers z=x+iyz=x+iy satisfying arg⁡(z+αz+β)=π4\arg\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}, the ordered pair (x,y)(x,y) lies on the circle x2+y2+5x−3y+4=0x^2+y^2+5x-3y+4=0 Then which of the following statements is (are) TRUE?
Not attempted
A
α=−1\alpha=-1
B
αβ=4\alpha\beta=4
C
αβ=−4\alpha\beta=-4
D
β=4\beta=4
14
For x∈Rx\in\mathbb{R}, the number of real roots of the equation 3x2−4∣x2−1∣+x−1=03x^2-4|x^2-1|+x-1=0 is ____.
Not attempted
Your answer: —
Correct answer: 4
15
Let θ1,θ2,…,θ10\theta_1,\theta_2,\ldots,\theta_{10} be positive valued angles (in radian) such that θ1+θ2+⋯+θ10=2π\theta_1+\theta_2+\cdots+\theta_{10}=2\pi. Define the complex numbers z1=eiθ1z_1=e^{i\theta_1}, zk=zk−1eiθkz_k=z_{k-1}e^{i\theta_k} for k=2,3,…,10k=2,3,\ldots,10, where i=−1i=\sqrt{-1}. Consider the statements PP and QQ given below: P:∣z2−z1∣+∣z3−z2∣+⋯+∣z10−z9∣+∣z1−z10∣≤2πP: |z_2-z_1|+|z_3-z_2|+\cdots+|z_{10}-z_9|+|z_1-z_{10}| \le 2\pi Q:∣z22−z12∣+∣z32−z22∣+⋯+∣z102−z92∣+∣z12−z102∣≤4πQ: |z_2^2-z_1^2|+|z_3^2-z_2^2|+\cdots+|z_{10}^2-z_9^2|+|z_1^2-z_{10}^2| \le 4\pi Then,
Not attempted
A
PP is TRUE and QQ is FALSE
B
QQ is TRUE and PP is FALSE
C
both PP and QQ are TRUE
D
both PP and QQ are FALSE
16
For a complex number zz, let Re(z)\text{Re}(z) denote the real part of zz. Let SS be the set of all complex numbers zz satisfying z4−∣z∣4=4iz2z^4 - |z|^4 = 4iz^2, where i=−1i=\sqrt{-1}. Then the minimum possible value of ∣z1−z2∣2|z_1-z_2|^2, where z1,z2∈Sz_1, z_2 \in S with Re(z1)>0\text{Re}(z_1) > 0 and Re(z2)<0\text{Re}(z_2) < 0, is ____
Not attempted
Your answer: —
Correct answer: 8
17
Let SS be the set of all complex numbers zz satisfying ∣z2+z+1∣=1|z^2+z+1|=1. Then which of the following statements is/are TRUE?
Not attempted
A
∣z+12∣≤12|z+\frac{1}{2}| \leq \frac{1}{2} for all z∈Sz \in S
B
∣z∣≤2|z| \leq 2 for all z∈Sz \in S
C
∣z+12∣≥12|z+\frac{1}{2}| \geq \frac{1}{2} for all z∈Sz \in S
D
The set SS has exactly four elements
18
Suppose a,ba, b denote the distinct real roots of the quadratic polynomial x2+20x−2020x^2 + 20x - 2020, and suppose c,dc, d denote the distinct complex roots of the quadratic polynomial x2−20x+2020x^2 - 20x + 2020. Then the value of ac(a−c)+ad(a−d)+bc(b−c)+bd(b−d)ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) is
Not attempted
A
00
B
80008000
C
80808080
D
1600016000
19
Let SS be the set of all complex numbers zz satisfying ∣z−2+i∣≥5|z-2+i|\ge\sqrt5. If the complex number z0z_0 is such that 1∣z0−1∣\dfrac{1}{|z_0-1|} is the maximum of the set {1∣z−1∣:z∈S}\left\{\dfrac{1}{|z-1|}:z\in S\right\}, then the principal argument of 4−z0−zˉ0z0−zˉ0+2i\dfrac{4-z_0-\bar z_0}{z_0-\bar z_0+2i} is
Not attempted
A
π4\dfrac{\pi}{4}
B
−π2-\dfrac{\pi}{2}
C
3π4\dfrac{3\pi}{4}
D
π2\dfrac{\pi}{2}
20
Let ω≠1\omega\ne1 be a cube root of unity. Then the minimum of the set {∣a+bω+cω2∣2:a,b,c distinct non-zero integers}\{|a+b\omega+c\omega^2|^2 : a,b,c\text{ distinct non-zero integers}\} equals _____
Not attempted
Your answer: —
Correct answer: 3
21
Let s,t,rs,t,r be non-zero complex numbers and LL be the set of solutions z=x+iyz=x+iy (x,y∈R,i=−1)(x,y\in\mathbb{R}, i=\sqrt{-1}) of the equation sz+tzˉ+r=0sz+t\bar z+r=0, where zˉ=x−iy\bar z=x-iy. Then, which of the following statement(s) is (are) TRUE?
Not attempted
A
If LL has exactly one element, then ∣s∣≠∣t∣|s|\neq|t|
B
If ∣s∣=∣t∣|s|=|t|, then LL has infinitely many elements
C
The number of elements in L∩{z:∣z−1+i∣=5}L\cap\{z:|z-1+i|=5\} is at most 22
D
If LL has more than one element, then LL has infinitely many elements
22
For a non-zero complex number zz, let arg⁡(z)\arg(z) denote the principal argument with −π<arg⁡(z)≤π-\pi < \arg(z) \le \pi. Then, which of the following statement(s) is (are) FALSE?
Not attempted
A
arg⁡(−1−i)=π4\arg(-1-i) = \dfrac{\pi}{4}, where i=−1i=\sqrt{-1}
B
The function f:R→(−π,π]f:\mathbb{R}\to(-\pi,\pi], defined by f(t)=arg⁡(−1+it)f(t)=\arg(-1+it) for all t∈Rt\in\mathbb{R}, is continuous at all points of R\mathbb{R}, where i=−1i=\sqrt{-1}
C
For any two non-zero complex numbers z1z_1 and z2z_2, arg⁡(z1z2)−arg⁡(z1)+arg⁡(z2)\arg\left(\dfrac{z_1}{z_2}\right)-\arg(z_1)+\arg(z_2) is an integer multiple of 2π2\pi
D
For any three given distinct complex numbers z1,z2,z3z_1,z_2,z_3, the locus of the point zz satisfying arg⁡((z−z1)(z2−z3)(z−z3)(z2−z1))=π\arg\left(\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right)=\pi lies on a straight line
23
Let a,b,xa, b, x and yy be real numbers such that a−b=1a-b=1 and y≠0y\neq0. If the complex number z=x+iyz=x+iy satisfies Im(az+bz+1)=y\text{Im}\left(\dfrac{az+b}{z+1}\right)=y, then which of the following is(are) possible value(s) of xx?
Not attempted
A
−1+1−y2-1+\sqrt{1-y^2}
B
−1−1−y2-1-\sqrt{1-y^2}
C
1+1+y21+\sqrt{1+y^2}
D
1−1+y21-\sqrt{1+y^2}
24
Let a,b∈Ra,b\in\mathbb{R} and a2+b2≠0a^2+b^2\ne0. Suppose S={z∈C:z=1a+ibt, t∈R, t≠0}S=\left\{z\in\mathbb{C}: z=\dfrac{1}{a+ibt},\ t\in\mathbb{R},\ t\ne0\right\}, where i=−1i=\sqrt{-1}. If z=x+iyz=x+iy and z∈Sz\in S, then (x,y)(x,y) lies on
Not attempted
A
the circle with radius 12a\dfrac{1}{2a} and centre (12a,0)\left(\dfrac{1}{2a},0\right) for a>0,b≠0a>0,b\ne0
B
the circle with radius −12a-\dfrac{1}{2a} and centre (−12a,0)\left(-\dfrac{1}{2a},0\right) for a<0,b≠0a<0,b\ne0
C
the xx-axis for a≠0,b=0a\ne0,b=0
D
the yy-axis for a=0,b≠0a=0,b\ne0
25
Let z=−1+3i2z=\dfrac{-1+\sqrt3i}{2}, where i=−1i=\sqrt{-1}, and r,s∈{1,2,3}r,s\in\{1,2,3\}. Let P=((−z)rz2sz2szr)P=\begin{pmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{pmatrix} and II be the identity matrix of order 2. Then the total number of ordered pairs (r,s)(r,s) for which P2=−IP^2=-I is
Not attempted
Your answer: —
Correct answer: 1
26
Let SS be the set of all non-zero real numbers α\alpha such that the quadratic equation αx2−x+α=0\alpha x^2-x+\alpha=0 has two distinct real roots x1x_1 and x2x_2 satisfying the inequality ∣x1−x2∣<1|x_1-x_2|<1. Which of the following intervals is(are) a subset(s) of SS?
Not attempted
A
(−12,−15)\left(-\dfrac12,-\dfrac1{\sqrt5}\right)
B
(−15,0)\left(-\dfrac1{\sqrt5},0\right)
C
(0,15)\left(0,\dfrac1{\sqrt5}\right)
D
(15,12)\left(\dfrac1{\sqrt5},\dfrac12\right)
27
For any integer kk, let αk=cos⁡(kπ7)+isin⁡(kπ7)\alpha_k=\cos\left(\dfrac{k\pi}{7}\right)+i\sin\left(\dfrac{k\pi}{7}\right), where i=−1i=\sqrt{-1}. The value of the expression ∑k=112∣αk+1−αk∣∑k=13∣α4k−1−α4k−2∣\dfrac{\displaystyle\sum_{k=1}^{12}\left|\alpha_{k+1}-\alpha_k\right|}{\displaystyle\sum_{k=1}^{3}\left|\alpha_{4k-1}-\alpha_{4k-2}\right|} is
Not attempted
Your answer: —
Correct answer: 4
28
This question is part of a Column-I to Column-II matching set (JEE Advanced 2015 Section 3, Q.59). Column II lists the values 1,2,3,4,51,2,3,4,5; select every value that correctly matches this Column-I entry. Entry (C)\textbf{Entry (C)}: Let ω≠1\omega\ne1 be a complex cube root of unity. If (3−3ω+2ω2)4n+3+(2+3ω−3ω2)4n+3+(−3+2ω+3ω2)4n+3=0(3-3\omega+2\omega^2)^{4n+3}+(2+3\omega-3\omega^2)^{4n+3}+(-3+2\omega+3\omega^2)^{4n+3}=0, then possible value(s) of nn is (are)
Not attempted
A
11
B
22
C
33
D
44
E
55
29
Let zk=cos⁡(2kπ10)+isin⁡(2kπ10)z_k=\cos\left(\dfrac{2k\pi}{10}\right)+i\sin\left(\dfrac{2k\pi}{10}\right), k=1,2,…,9k=1,2,\dots,9 (the nonzero-index 10th10^{\text{th}} roots of unity). Match each entry of List I\textbf{List I} to its value in List II\textbf{List II}. List I\textbf{List I} P.\textbf{P.} For each zkz_k there exists a zjz_j such that zk⋅zj=1z_k\cdot z_j=1. Q.\textbf{Q.} There exists a k∈{1,2,…,9}k\in\{1,2,\dots,9\} such that z1⋅z=zkz_1\cdot z=z_k has no solution zz in the set of complex numbers. R.\textbf{R.} ∣1−z1∣∣1−z2∣⋯∣1−z9∣10\dfrac{|1-z_1||1-z_2|\cdots|1-z_9|}{10} equals S.\textbf{S.} 1−∑k=19cos⁡(2kπ10)1-\displaystyle\sum_{k=1}^{9}\cos\left(\dfrac{2k\pi}{10}\right) equals List II\textbf{List II} 1.\textbf{1.} True 2.\textbf{2.} False 3.\textbf{3.} 11 4.\textbf{4.} 22 Which of the following is the correct combination?
Not attempted
A
P→1, Q→2, R→4, S→3P{\to}1,\ Q{\to}2,\ R{\to}4,\ S{\to}3
B
P→2, Q→1, R→3, S→4P{\to}2,\ Q{\to}1,\ R{\to}3,\ S{\to}4
C
P→1, Q→2, R→3, S→4P{\to}1,\ Q{\to}2,\ R{\to}3,\ S{\to}4
D
P→2, Q→1, R→4, S→3P{\to}2,\ Q{\to}1,\ R{\to}4,\ S{\to}3
30
The quadratic equation p(x)=0p(x)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x))=0p(p(x))=0 has
Not attempted
A
Only purely imaginary roots
B
All real roots
C
Two real and two purely imaginary roots
D
Neither real nor purely imaginary roots
31
Let w=3+i2w=\dfrac{\sqrt3+i}{2} and P={wn:n=1,2,3,…}P=\{w^n: n=1,2,3,\ldots\}. Further H1={z∈C:Re⁡z>12}H_1=\left\{z\in\mathbb{C}: \operatorname{Re} z>\dfrac12\right\} and H2={z∈C:Re⁡z<−12}H_2=\left\{z\in\mathbb{C}: \operatorname{Re} z<-\dfrac12\right\}, where C\mathbb{C} is the set of all complex numbers. If z1∈P∩H1z_1\in P\cap H_1, z2∈P∩H2z_2\in P\cap H_2 and OO represents the origin, then ∠z1Oz2=\angle z_1Oz_2=
Not attempted
A
π2\dfrac{\pi}{2}
B
π6\dfrac{\pi}{6}
C
2π3\dfrac{2\pi}{3}
D
5π6\dfrac{5\pi}{6}
32
If 3x=4x−13^x=4^{x-1}, then x=x=
Not attempted
A
2log⁡322log⁡32−1\dfrac{2\log_3 2}{2\log_3 2-1}
B
22−log⁡23\dfrac{2}{2-\log_2 3}
C
11−log⁡43\dfrac{1}{1-\log_4 3}
D
2log⁡232log⁡23−1\dfrac{2\log_2 3}{2\log_2 3-1}
33
Let S=S1∩S2∩S3S=S_1\cap S_2\cap S_3, where S1={z∈C:∣z∣<4}S_1=\{z\in\mathbb{C}: |z|<4\}, S2={z∈C:Im⁡[z−1+3 i1−3 i]>0}S_2=\left\{z\in\mathbb{C}: \operatorname{Im}\left[\dfrac{z-1+\sqrt3\,i}{1-\sqrt3\,i}\right]>0\right\} and S3={z∈C:Re⁡ z>0}S_3=\{z\in\mathbb{C}: \operatorname{Re}\,z>0\}. Area of S=S=
Not attempted
A
10π3\dfrac{10\pi}{3}
B
20π3\dfrac{20\pi}{3}
C
16π3\dfrac{16\pi}{3}
D
32π3\dfrac{32\pi}{3}
34
Let S=S1∩S2∩S3S=S_1\cap S_2\cap S_3, where S1={z∈C:∣z∣<4}S_1=\{z\in\mathbb{C}: |z|<4\}, S2={z∈C:Im⁡[z−1+3 i1−3 i]>0}S_2=\left\{z\in\mathbb{C}: \operatorname{Im}\left[\dfrac{z-1+\sqrt3\,i}{1-\sqrt3\,i}\right]>0\right\} and S3={z∈C:Re⁡ z>0}S_3=\{z\in\mathbb{C}: \operatorname{Re}\,z>0\}. min⁡z∈S∣1−3i−z∣=\displaystyle\min_{z\in S}|1-3i-z|=
Not attempted
A
2−32\dfrac{2-\sqrt3}{2}
B
2+32\dfrac{2+\sqrt3}{2}
C
3−32\dfrac{3-\sqrt3}{2}
D
3+32\dfrac{3+\sqrt3}{2}
35
Let complex numbers α\alpha and 1α\dfrac{1}{\alpha} lie on circles (x−x0)2+(y−y0)2=r2(x-x_0)^2+(y-y_0)^2=r^2 and (x−x0)2+(y−y0)2=4r2(x-x_0)^2+(y-y_0)^2=4r^2 respectively. If z0=x0+iy0z_0=x_0+iy_0 satisfies 2∣z0∣2=r2+22|z_0|^2=r^2+2, then ∣α∣=|\alpha|=
Not attempted
A
12\dfrac{1}{\sqrt2}
B
12\dfrac{1}{2}
C
17\dfrac{1}{\sqrt7}
D
13\dfrac{1}{3}
Write both circle conditions as ∣α−z0∣2=r2|\alpha-z_0|^2=r^2 and ∣1/α−z0∣2=4r2|1/\alpha - z_0|^2=4r^2, expand using Re\text{Re}, and exploit rotational symmetry: since rotating α\alpha and z0z_0 by the same angle preserves both circle equations and ∣z0∣|z_0|, you may choose z0z_0 to be real without loss of generality.
Deconstruction\textbf{Deconstruction}: We need ∣α∣|\alpha| given two circle-membership conditions on α\alpha and 1/α1/\alpha, linked through z0z_0 by 2∣z0∣2=r2+22|z_0|^2=r^2+2. Strategy\textbf{Strategy}: Expand ∣α−z0∣2=r2|\alpha-z_0|^2=r^2 and ∣1/α−z0∣2=4r2|1/\alpha-z_0|^2=4r^2 in terms of Re(αzˉ0)\text{Re}(\alpha\bar z_0) and Re(αz0)\text{Re}(\alpha z_0). The whole configuration is invariant under simultaneously rotating α\alpha and z0z_0 by the same angle (this preserves both circle equations and ∣z0∣|z_0|, hence ∣α∣|\alpha| is unaffected), so we may choose z0z_0 real, which forces Re(αzˉ0)=Re(αz0)\text{Re}(\alpha\bar z_0)=\text{Re}(\alpha z_0) and collapses the system to one variable. Execution\textbf{Execution}: Write α=p+iq\alpha=p+iq. Circle 1 gives ∣α∣2−2Re(αzˉ0)+∣z0∣2=r2|\alpha|^2-2\text{Re}(\alpha\bar z_0)+|z_0|^2=r^2. Circle 2 (on 1/α1/\alpha) gives 1∣α∣2−2Re(αz0)∣α∣2+∣z0∣2=4r2\dfrac{1}{|\alpha|^2}-\dfrac{2\text{Re}(\alpha z_0)}{|\alpha|^2}+|z_0|^2=4r^2. - Choose z0=x0z_0=x_0 real (WLOG by rotation). Then Re(αzˉ0)=Re(αz0)=px0=:A\text{Re}(\alpha\bar z_0)=\text{Re}(\alpha z_0)=p x_0=:A. - Let s=∣α∣2s=|\alpha|^2. Eq.(1): A=s+x02−r2A=s+x_0^2-r^2. - Eq.(2): 1−As=4r2−x02⇒1−A=s(4r2−x02)\dfrac{1-A}{s}=4r^2-x_0^2 \Rightarrow 1-A=s(4r^2-x_0^2). - Substituting AA: 1−x02+r2=s(1−x02+4r2)1-x_0^2+r^2=s(1-x_0^2+4r^2), so s=1−x02+r21−x02+4r2s=\dfrac{1-x_0^2+r^2}{1-x_0^2+4r^2}. - Given constraint (with z0z_0 real, ∣z0∣2=x02|z_0|^2=x_0^2): 2x02=r2+2⇒x02=r2+222x_0^2=r^2+2 \Rightarrow x_0^2=\dfrac{r^2+2}{2}. - Then 1−x02+r2=1−r2+22+r2=r221-x_0^2+r^2 = 1-\dfrac{r^2+2}{2}+r^2=\dfrac{r^2}{2}, and 1−x02+4r2=1−r2+22+4r2=7r221-x_0^2+4r^2=1-\dfrac{r^2+2}{2}+4r^2=\dfrac{7r^2}{2}. - So s=r2/27r2/2=17s=\dfrac{r^2/2}{7r^2/2}=\dfrac{1}{7}, giving ∣α∣=17|\alpha|=\dfrac{1}{\sqrt7}. Verification\textbf{Verification}: The result is independent of rr (it cancels), consistent with the problem expecting a numeric answer purely from the given relation 2∣z0∣2=r2+22|z_0|^2=r^2+2 — a strong consistency check that the algebra is dimensionally sound. Final Answer\textbf{Final Answer}: (C) 17\dfrac{1}{\sqrt7}