Question

Match the colums:

$\begin{array}{cc}Column1& Column2\\ \text{a. The set of points}z\text{satisfying}|z-i|z||=|z+i|z||& \text{p. an ellipse with eccentricity}\frac{4}{5}\\ \text{is contained in or equal to}& \\ \text{b. The set of points}z\text{satisfying}|z+4|+|z-4|=10& \text{q. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& \text{Img}z=0\\ \text{c. If}|w|=2,\text{then the set of points}z=w-\frac{1}{w}& \text{r. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Img}z|\le 10\\ \text{d. If}|w|=1,\text{then the set of points}z=w+\frac{1}{w}& \text{s. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Re}z|<2\\ & \text{t. the set of points}z\text{satisfying}\\ & |z|\le 3\end{array}$

• A. $a-q,r;b-p;c-p,s,t;d-q,r,t$
• B. $a-p,q,r;b-p;c-s,t;d-q,r,t$
• C. $a-q,r;b-p,t;c-p,t;d-q,r$
• D. $a-p,r;b-p,q;c-s,t;d-p,r$

Answer 1

The correct option is A $a-q,r;b-p;c-p,s,t;d-q,r,t$

$\text{a.}$

$|\frac{z}{|z|}-i|=|\frac{z}{|z|}+i|,z\ne 0$
$\frac{z}{|z|}$ is unimodular complex number and lies on perpendicular bisector of $i$ and $-i$.
$\frac{z}{|z|}=\pm 1\Rightarrow z=\pm |z|$
$\Rightarrow z$ is a real number
$\Rightarrow \text{Im}z=0$.

$\text{b.}$

$|z+4|+|z-4|=10$
$z$ lies on an ellipse whose focus are $(4,0)$ and $(-4,0)$ and length of major axis is $10$
$\Rightarrow 2ae=8$ and $2a=10\Rightarrow e=\frac{4}{5}$
$|\text{Re}z|\le 5$

$\text{c.}$

$|w|=2\Rightarrow w=2(\cos \theta +i\sin \theta )$
$\Rightarrow z=x+iy=2(\cos \theta +i\sin \theta )-\frac{1}{2}(\cos \theta -i\sin \theta )$
$\Rightarrow z=\frac{3}{2}\cos \theta +i\frac{5}{2}\sin \theta$
$\Rightarrow \frac{x^2}{(3/2{)}^2}+\frac{y^2}{(5/2{)}^2}$
$\Rightarrow e=1-\frac{9/4}{25/4}=\frac{4}{5}$

$\text{d.}$
$|w|=1\Rightarrow x+iy=\cos \theta +i\sin \theta +\cos \theta -i\sin \theta$
$\Rightarrow x+iy=2\cos \theta$
$|\text{Re}\left(z\right)|\le 1,\text{Im}\left(z\right)=0$