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. 6
• B. $6\pm 8i$
• C. 6+8i, 6+17i
• D. None of these

Answer 1

The correct option is C

6+8i, 6+17i

We have $\left|\frac{z-12}{z-8i}\right|$=$\frac{5}{3}$, $\left|\frac{z-4}{z-8}\right|$=1

Let z=x+iy, then

$\left|\frac{z-12}{z-8i}\right|$=$\frac{5}{3}$ $\Rightarrow$ 3|z-12| = 5|z-8i|

$\Rightarrow$ $9(x-12{)}^2$+$9y^2$ = 25$x^2$+25$(y-8{)}^2$ $\cdots$$\cdots$(i)

$\left|\frac{z-4}{z-8}\right|$=1 $\Rightarrow$ |z-4| = |z-8|

$\Rightarrow$ $(x-4{)}^2$+$y^2$ = $y^2$+$(x-8{)}^2$ $\Rightarrow$ x=6

Putting x=6 in (i), we get $y^2$-25y+136=0

y=17, 8

Hence z=6+17i or z=6+8i

Trick: Check it with options.