Question

If $z_1=2+3i$ and $z_2=3+4i$ be two points on the complex plane. Then, the set of complex number $z$ is 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 or straight line

Answer 1

Answer: (C)

a circle

Given $z_1=2+3i$ and $z_2=3+4i$
Now we have,
${|z-z_1|}^2+{|z-z_2|}^2={|z_1-z_2|}^2$

$\Rightarrow {\left|\right(x+iy)-(2-3i\left)\right|}^2+{\left|\right(x+iy)-(3+4i\left)\right|}^2={\left|\right(2+3i)-(3-4i\left)\right|}^2$

$\Rightarrow {\left|\right(x-2+i(y-3)\left)\right|}^2+{\left|\right(x-3)+i(y-4\left)\right|}^2={|-1-i|}^2$

$\Rightarrow {(x-2)}^2+{(y-3)}^2+{(x-3)}^2+{(y-4)}^2=1+1$

$\Rightarrow$ $x^2+4-4x+y^2+9-16y+x^2+9-6x+y^2+!6-8y=2$

$\Rightarrow$ $2x^2+2y^2-10x-14y+36=0$

$\Rightarrow$ $x^2+y^2-5x-7y+18=0$
which represent a circle with centre $(\frac{5}{2},\frac{7}{2})$ and radius $\sqrt{\frac{25}{4}+\frac{49}{4}-18}=\frac{1}{\sqrt{2}}$