Question

Match the statements in column-I with those in column-II.
[Note: Here $z$ takes the values in the complex plane and $\text{Im}z$ and $\text{Re}z$ denote, respectively, the imaginary part and the real part of $z$ ]

Column I		Column II
A	The set of points $z$ satisfying $|z-i|z||=|z+i|z||$ is contained in or equal to	p	an ellipse with eccentricity $\frac{4}{5}$
B	The set of points $z$ satisfying $|z+4|+|z-4|=10$	q	the set of points z satisfying Im $z=0$ is contained in or equal to
C	If $|\omega |=2$, then the set of points $z=\omega -1/\omega$ is	r	the set of points $z$ satisfying $|\text{Im}z|\le 1$ contained in or equal to
D	If $|\omega |=1$, then the set of points $z=\omega +1/\omega$ is	s	the set of points $z$ satisfying $|\text{Re}z|\le 1$ contained in or equal to
		t	the set of points $z$ satisfying $|z|\le 3$

• A. $(A)\to (Q);(B)\to (P);(C)\to (P)(T);(D)\to (Q),(T)$
• B. $(A)\to (Q);(B)\to (Q);(C)\to (P)(T);(D)\to (Q),(S)$
• C. $(A)\to (P);(B)\to (P);(C)\to (P)(T);(D)\to (Q),(T)$
• D. $(A)\to (R);(B)\to (P);(C)\to (R)(T);(D)\to (Q),(T)$

Answer 1

Answer: (A)

(A)$(q)$
$\left|\frac{z}{|z|}-i\right|=\left|\frac{z}{|z|}+i\right|,z\ne 0$
$\frac{z}{|z|}$ is unimodular complex number
and lies on perpendicular bisector of $i$ and $-i$
$\Rightarrow \frac{z}{|z|}=\pm 1$
$\Rightarrow z=\pm 1|z|$
$\Rightarrow a$ is real number
$\Rightarrow \text{Im}(z)=0$
(B) $(p)$
$|z+4|+|z-4|=10$
$z$ lies on an ellipse whose focus are $(4,0)$ and $(-4,0)$ and length of major axis is $10$
$\Rightarrow 2ae=8$ and $2a=10$
$\Rightarrow e=4/5$
$|\text{Re}(z)|\le 5$
(C) $(p),(t)$
$|w|=2$
$\Rightarrow w=2(\cos \theta +i\sin \theta )$
$x+iy=2(\cos \theta +i\sin \theta )-\frac{1}{2}(\cos \theta -i\sin \theta )$
$=\frac{3}{2}\cos \theta +i\frac{5}{2}\sin \theta$
$\Rightarrow \frac{x^2}{(3/2{)}^2}+\frac{y^2}{(5/2{)}^2}=1$
$e^2=1-\frac{9/4}{25/4}=1-\frac{9}{25}=\frac{16}{25}$
$\Rightarrow e=\frac{4}{5}$
(D)$(q),(t)$
$|w|=1$
$\Rightarrow x+iy=\cos +i\sin \theta +\cos \theta -i\sin \theta$
$x+iy=2\cos \theta$
$|\text{Re}(z)|\le 1,\mid \text{Im}(z)=0$