Question

Let $z_1=2+3i$ and $z_2=3+4i$ be two points on the complex plane. Then the set of complex numbers $z$ satisfying $|z-z_1{|}^2+|z-z_2{|}^2=|z_1-z^2{|}^2$ represents

• A. a straight line
• B. a point
• C. a circle
• D. a pair of straight lines

Answer 1

Answer: (C)

Given, $z_1=2+3i$ and $z_2=3+4i$
Now, we have
${\left|z-z_1\right|}^2+{\left|z-z_2\right|}^2={\left|z_1-z_2\right|}^2$ (let $z=x+iy)$
$\Rightarrow |(x+iy)-(2+3i){|}^2+|(x+iy)-(3+4i){|}^2$
$=|(2+3i)-(3+4i){|}^2$
$\Rightarrow |(x-2)+i(y-3){|}^2+|(x-3)+i(y-4){|}^2$
$=|-1-i{|}^2$
$\Rightarrow (x-2{)}^2+(y-3{)}^2+(x-3{)}^2+(y-4{)}^2=1+1$
$\Rightarrow x^2+4-4x+y^2+9-6y$
$+x^2+9-6x+y^2+16-8y=2$
$\Rightarrow 2x^2+2y^2-10x-14y+36=0$
$\Rightarrow x^2+y^2-5x-7y+18=0$
which represent a circle with centre $\left(\frac{5}{2,}\frac{7}{2}\right)$ and
radius $\sqrt{\frac{25}{4}+\frac{49}{4}-18}=\frac{1}{\sqrt{2}}$