Question

Find the complex number z satisfying the equations $\left|\frac{z-12}{z-8i}\right|=\frac{5}{3},\left|\frac{z-4}{z-8}\right|=1$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$wehave\left|\frac{z-12}{z-8i}\right|=\frac{5}{3}and\left|\frac{z-4}{z-8}\right|=1letz=x+iy,then\left|\frac{z-12}{z-8i}\right|=\frac{5}{3}\Rightarrow 3|z-12|=5|z-8i|\Rightarrow 3|(x-12)+iy|=5|x+(y-8)i|\Rightarrow 9(x-12{)}^2+9y^2=25x^2+25(y-8{)}^2.........\left(i\right)$
$and\left|\frac{z-4}{z-8}\right|=1\Rightarrow |z-4|=|z-8|\Rightarrow |x-4+iy|=|z-4|=|z-8|\Rightarrow (x-4{)}^2+y^2=(x-8{)}^2+y^2\Rightarrow x=6Putingx=6in\left(i\right),weget{y}^2-25y+136=0\therefore y=17,8Hencez=6+17iorz=6+8i$
Trick: Check it with options