Question

Let $P\left(z\right)$ be a point in complex plane satisfying $z\overline{z}+(4-5i)\overline{z}+(4+5i)z=40.$ If $a=\text{max}|z+2-3i|$ and $b=\text{min}|z+2-3i|,$ then

• A. $a+b=18$
• B. $a+b=9$
• C. $a-b=4\sqrt{2}$
• D. $ab=73$

Answer 1

Answer: (A), (C), (D)

$ab=73$

General equation of circle in complex form is $z\overline{z}+a\overline{z}+\overline{a}z+b=0,$
where centre is $-a$ and radius is $\sqrt{a\overline{a}-b}$
So, centre of circle is $C(-4,5)$
and radius, $r=\sqrt{(-4{)}^2+(5{)}^2+40}=9$

Distance of centre $C(-4,5)$ from $P(-2,3)$ is $2\sqrt{2}<r=9$
So, $-2+3i$ lies inside the circle.


$\therefore a=\text{max}|z-(-2+3i\left)\right|=r+CP=9+2\sqrt{2}$
and $b=\text{min}|z-(-2+3i\left)\right|=r-CP=9-2\sqrt{2}$

Hence, $a+b=18$
$a-b=4\sqrt{2}$
and $ab=(9+2\sqrt{2})(9-2\sqrt{2})=81-8=73$