Question

Find the complex number $z$ satisfying the equation
$\left|\frac{z-4}{z-8}\right|=1$ and $|z|=10$ .

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

Answer 1

The correct option is B $6\pm 8i$

Let $z=x+iy$

Now,

$|\frac{z-4}{z-8}|=1⟹\frac{|z-4|}{|z-8|}=1$

$⟹|z-4|=|z-8|$

$⟹|(x-4)+iy|=|(x-8)+iy|$...........[putting $z=x+iy$]

$⟹\sqrt{(x-4{)}^2+y^2}=\sqrt{(x-8{)}^2+y^2}$

Squaring both sides:

$⟹x^2-8x+16+y^2=x^2-16x+64+y^2$

$⟹8x=48⟹x=6$..........$\left(1\right)$

Also $|z|=10⟹\sqrt{x^2+y^2}=10$

$⟹x^2+y^2=100⟹y^2=100-x^2$

$⟹y^2=100-36$..............[$x=6$;so $x^2=36$ from eq. (1)]

$⟹y=\sqrt{64}⟹y=\pm 8$

So, $z=6\pm i8$