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 $|\text{Im} z| \leq 1$ 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 $|\text{Re} z| \leq 1$ contained in or equal to
		t	the set of points $z$ satisfying $|z| \leq 3$

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

Answer 1

Answer: (A)

(A)$(q)$
$\left|\frac{z}{|z|}-i\right|=\left|\frac{z}{|z|}+i\right|, z \neq 0$
$\frac{z}{|z|}$ is unimodular complex number
and lies on perpendicular bisector of $i$ and $-i$
$\Rightarrow \frac{z}{|z|}=\pm 1 $
$\Rightarrow 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 $10$
$\Rightarrow 2 ae =8$ and $2 a =10$
$ \Rightarrow e =4 / 5$
$|\text{Re}(z)| \leq 5$
(C) $(p),(t) $
$|w|=2 $
$\Rightarrow w=2(\cos \theta+i \sin \theta) $
$x+i y=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 $
$\Rightarrow x+i y=\cos +i \sin \theta+\cos \theta-i \sin \theta$
$x+i y=2 \cos \theta$
$|\text{Re}(z)| \leq 1, \mid \text{Im}(z)=0$