Question

If $z_{1}and{z}_2$ are lying on $|z-3|\le 4$ and $|z-1|+|z+1|=3$ respectively then range of $|z_1-z_2|$ is

• A. [0,
• B. [0, 1]
• C.
• D.

Answer 1

The correct option is C

$|z-3|\le 4$represents circle and its interior whose centre is (3, 0) and radius 4.
$|z-1|+|z+1|=3$ represents set of all points on the boundary of ellipse whose foci at S(1, 0) and S' (-1, 0) vertices A = $(\frac{3}{2},0)$, A' = $(\frac{-3}{2},0)$.
Minimum value of$|z_1-z_2|$ is 0 when $z_{1}and{z}_2$ coincide at the intersection of circle and ellipse. $|z_1-z_2|$is Maximum if $z_1$ is at (7, 0) on the circle and $z_2$ at A' $(\frac{-3}{2},0)$ on the ellipse
$\therefore Maxvalueof|z_1-z_2|$ is 7 + $\frac{3}{2}=\frac{17}{2}$
$\therefore$ Range of $|z_1-z_2|$ is $[0,\frac{17}{2}]$