Question

Match the statements in Column-I with those in Column-II.
[Note: Here $z$ takes values in the complex plane and $\text{Im}z$ and $\text{Re}z$ denote, respectively, the imaginary part and the real part of $z$.]
$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(A)}& \text{The set of points}z\text{satisfying}& \text{(p)}& \text{An ellipse with}\\ & |z-i|z||=|z+i|z||\text{is}& & \text{eccentricity}\frac{4}{5}\\ & \text{contained in or equal to}& & \\ \text{(B)}& \text{The set of points}z\text{satisfying}& \text{(q)}& \text{The set of points}z\\ & |z+4|+|z-4|=10\text{is}& & \text{satisfying Im}z=0\\ & \text{contained in or equal to}& & \\ \text{(C)}& \text{If}|w|=2,\text{then the set of}& \text{(r)}& \text{The set of points}z\\ & \text{points}z=w-\frac{1}{w}\text{is}& & \text{satisfying}|\text{Im}z|\\ & \text{contained in or equal to}& & \le 1\\ \text{(D)}& \text{If}|w|=1,\text{then the set of}& \text{(r)}& \text{The set of points}z\\ & \text{points}z=w-\frac{1}{w}\text{is}& & \text{satisfying}|\text{Re}z|\\ & \text{contained in or equal to}& & \le 2\\ & & \text{(t)}& \text{The set of points}z\\ & & & \text{satisfying}|z|\le 3\end{array}$

• A. Option c
• B. A-Q,R; B-P; C-P,S,T; D-Q,R,S,T
• C. Option d
• D. Option b

Answer 1

The correct option is B A-Q,R; B-P; C-P,S,T; D-Q,R,S,T

(A) $z$ is equidistant from the points $i|z|$ and $–i|z|$, whose perpendicular bisector is $\text{Im}\left(z\right)=0$.
(B) Sum of distance of $z$ from $(4,0)$ and $(–4,0)$ is a constant $10$, hence locus of $z$ is ellipse with semi-major axis $5$ and focus at $(\pm 4,0)$.
$ae=4$
$\therefore e=\frac{4}{5}$
(C) $|z|\le |w|+\left|\frac{1}{w}\right|=\frac{5}{2}<3$
(D) $|z|\le |w|+\left|\frac{1}{w}\right|=2$
$\therefore \text{Re}z\le |z|\le 2$