Question

The equation $z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4$ =0 represents a circle of radius

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

The correct option is B

3

Let the equation of the circle be $|$z - $z_1$$|$ = r, where $z_1$ is the centre and r,
the radius multiplying by conjugate, we get
$(z-z_1$)($\overline{z}$ - ${\overline{z}}_1$) = $r^2$
$z\overline{z}$ - ${\overline{z}}_{1}z-z_1\overline{z}+z_1{\overline{z}}_1-r^2$ = 0
Comparing it with
$z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4$, we get
-$z_1$ = $2+3i,z_1\overline{z}-r^2$ = $4$
$\Rightarrow$ $z_1$ = $-2-3i$
$z_1$${\overline{z}}_1$ - $r^2$ = $4$
$|$${\overline{z}}_1$$|$ - $r^2$ = $4$
$13-r^2$ = $4$
$r^2$ = $9$
$\Rightarrow$ r = $3$