Match the statements in column – I with those of column – II {Note : Here z takes the values in the complex plane and Imz and Re(z) denote, respectively, the imaginary part and the real part of z} Column – I Column – IIA)The set of points z satisfying z−iz−z+iz=0 is subset or equal toP)The set of points on or inside an ellipse with center at origin, length of major axis is 10 and eccentricity 45B)The set of points z satisfying |z+4|−|z−4|=0 is subset or equal toQ)The set of points z satisfying Im(z)=0C)If |ω|=2, then the set of points z=ω−(1/ω) is subset or equal toR)The set of points z satisfying |Im(z)|≤1D)If |ω|=1, then the set of points z=ω+1/ω is subset or equal toS)The set of points z satisfying |Re(z)|≤1 T)The set of points z satisfying |z|≤3The correct option is

a→q |z|z|−i|=|z|z|+i|,z≠0z|z| is unimodular complex numberAnd lies on perpendicular bisector of i and −i⇒z|z|±1⇒±1|z|⇒z is real number ⇒Lm(z)=0b→pZ lies on an ellipse whose focus are (4,0) and (−4,0) and length of major axis is 10⇒ 2ae=8 and 2a=10⇒e=45 |Re(z)|≤5 |ω|=2⇒ω=2(cosθ+isinθ) ⇒ω=x+iy=2(cosθ+isinθ)−12(cosθ−isinθ) =32cosθ+i52sinθ⇒x2(3/2)2+y2(5/2)2=1 ⇒ e2=1−9/125/4=1−925=1625⇒e=45 d→q,t |ω|=1⇒x+iy=cosθ+isinθ+cosθ−isinθ x+iy=2cosθ |Re(z)|≤1,Im(z)=0

Match the statements in column – I with those of column – II {Note : Here z takes the values in the complex plane and Imz and Re(z) denote, respectively, the imaginary part and the real part of z} Column – I Column – IIA)The set of points z satisfying z−iz−z+iz=0 is subset or equal toP)The set of points on or inside an ellipse with center at origin, length of major axis is 10 and eccentricity 45B)The set of points z satisfying |z+4|−|z−4|=0 is subset or equal toQ)The set of points z satisfying Im(z)=0C)If |ω|=2, then the set of points z=ω−(1/ω) is subset or equal toR)The set of points z satisfying |Im(z)|≤1D)If |ω|=1, then the set of points z=ω+1/ω is subset or equal toS)The set of points z satisfying |Re(z)|≤1 T)The set of points z satisfying |z|≤3The correct option is

a→q |z|z|−i|=|z|z|+i|,z≠0z|z| is unimodular complex numberAnd lies on perpendicular bisector of i and −i⇒z|z|±1⇒±1|z|⇒z is real number ⇒Lm(z)=0b→pZ lies on an ellipse whose focus are (4,0) and (−4,0) and length of major axis is 10⇒ 2ae=8 and 2a=10⇒e=45 |Re(z)|≤5 |ω|=2⇒ω=2(cosθ+isinθ) ⇒ω=x+iy=2(cosθ+isinθ)−12(cosθ−isinθ) =32cosθ+i52sinθ⇒x2(3/2)2+y2(5/2)2=1 ⇒ e2=1−9/125/4=1−925=1625⇒e=45 d→q,t |ω|=1⇒x+iy=cosθ+isinθ+cosθ−isinθ x+iy=2cosθ |Re(z)|≤1,Im(z)=0

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Match the statements in column – I with those of column – II {Note : Here z takes the values in the complex plane and $I m (z)$ and Re(z) denote, respectively, the imaginary part and the real part of z}
Column – I Column – II
A) The set of points z satisfying $|z - i |z|| - |z + i |z|| = 0$
is subset or equal to P) The set of points on or inside an ellipse with center at origin, length of major axis is 10 and eccentricity $\frac{4}{5}$
B) The set of points z satisfying $| z + 4 | - | z - 4 | = 0$ is subset or equal to Q) The set of points $z$ satisfying $Im (z) = 0$
C) If $| ω | = 2,$ then the set of points $z = ω - (1 / ω)$ is subset or equal to R) The set of points $z$ satisfying $| Im (z) | \leq 1$
D) If $| ω | = 1,$ then the set of points $z = ω + 1 / ω$ is subset or equal to S) The set of points $z$ satisfying $| Re (z) | \leq 1$
T) The set of points $z$ satisfying $| z | \leq 3$
The correct option is

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$A - Q R; B - S; C - P T; D - Q R T$

$A - R; B - S; C - P; D - T$

$A - Q; B - P; C - P T; D - Q T$

$A - Q R; B - S; C - P T; D - Q T$

answer is A.

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Detailed Solution

$a \to q | \frac{z}{| z |} - i | = | \frac{z}{| z |} + i |, z \neq 0$

$\frac{z}{| z |}$ is unimodular complex number

And lies on perpendicular bisector of $i$ and $- i$

$\Rightarrow \frac{z}{| z |} \pm 1 \Rightarrow \pm 1 | z |$

$\Rightarrow z$ is real number $\Rightarrow L m (z) = 0$

$b \to p$

Z lies on an ellipse whose focus are $(4, 0)$ and $(- 4, 0)$ and length of major axis is 10

$\Rightarrow 2 a e = 8 a n d 2 a = 10 \Rightarrow e = \frac{4}{5} | Re (z) | \leq 5 | ω | = 2 \Rightarrow ω = 2 (\cos θ + i \sin θ) \Rightarrow ω = x + i y = 2 (\cos θ + i \sin θ) - \frac{1}{2} (\cos θ - i \sin θ) = \frac{3}{2} \cos θ + i \frac{5}{2} \sin θ \Rightarrow \frac{x^{2}}{{(3 / 2)}^{2}} + \frac{y^{2}}{{(5 / 2)}^{2}} = 1 \Rightarrow e^{2} = 1 - \frac{9 / 1}{25 / 4} = 1 - \frac{9}{25} = \frac{16}{25} \Rightarrow e = \frac{4}{5} d \to q, t | ω | = 1 \Rightarrow x + i y = \cos θ + i \sin θ + \cos θ - i \sin θ x + i y = 2 \cos θ | Re (z) | \leq 1, Im (z) = 0$

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