Match the statements of Column I with these in Column II. [Note : Here $z$ takes values in the complex plane…
Match the statements of Column I with these in Column II.
[Note : Here $z$ takes values in the complex plane and $\operatorname{Im}(z)$ and $\operatorname{Re}(z)$ denote respectively, the imaginary part and the real part of $z$ ]
$\begin{array}{|c|c|c|c|}
\hline & \text { Column I } & & \text { Column II } \\
\hline \text { (A) } & \begin{array}{l}
\text { The set of points } z \text { satisfying } \\
|z-i||z|=|z+i||z| \text { is contained in or equal to }
\end{array} & \text { (p) } & \text { an ellipse with eccentricity } \frac{4}{5} \\
\hline \text { (B) } & \begin{array}{l}
\text { The set of points } z \text { satisfying } \\
|z+4|+|z-4|=10 \text { is contained in or equal to }
\end{array} & \text { (q) } & \begin{array}{l}
\text { the set of points } z \text { satisfying } \\
\operatorname{Im}(z)=0
\end{array} \\
\hline \text { (C) } & \begin{array}{l}
\text { If }|w|=2 \text {, then the set of points } z=w-\frac{1}{w} \text { is } \\
\text { contained in or equal to }
\end{array} & \text { (r) } & \begin{array}{l}
\text { the set of points } z \text { satisfying } \\
|\operatorname{Im}(z)| \leq 1
\end{array} \\
\hline \text { (D) } & \begin{array}{l}
\text { If }|w|=1 \text {, then the set of points } z=w+\frac{1}{w} \text { is } \\
\text { contained in or equal to }
\end{array} & \text { (s) } & \begin{array}{l}
\text { the set of points satisfying } \\
|\operatorname{Re}(z)| \leq 2
\end{array} \\
\hline & & (\mathrm{t}) & \begin{array}{l}
\text { the set of points } z \text { satisfying } \\
|z| \leq 3
\end{array} \\
\hline
\end{array}$
(A) q, (B) p, (C) p,s, (D) q,r,s
(A) q,r, (B) p, (C) p,s,t, (D) q,r,s,t
(A) q, (B) q, (C) p,s,t, (D) q,r,s,t
(A) q,r, (B) q, (C) p,s, (D) q,r,s
Solution
(A) $z$ is equidistant from the points $i|z|$ and $-i|z|$, whose perpendicular bisector is $\operatorname{Im}(z)=0$.
(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 $(\pm 4,0), a e=4$.
$
\therefore \quad e=\frac{4}{5}
$
(C) $|z| \leq|w|+\left|\frac{1}{w}\right|=\frac{5}{2} < 3$
(D) $|z| \leq|w|+\left|\frac{1}{w}\right|=2$
$
\therefore \quad \operatorname{Re}(z) \leq|z| \leq 2
$