Question

A function $f$ is defined on the complex number by $f( z )=( a + b i) z$, where ' $a$ ' and 'b' are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $| a + b i|=8$ and that $b ^2=\frac{ u }{ v }$ where $u$ and $v$ are coprimes. Find the value of $(u+v)$.

Answer 1

Given $|( a + bi ) z - z |=|( a + bi ) z | $
$| z ( a -1)+ b i z |=| az + bzi | $
$| z ||( a -1)+ b i|=| z || a + b i| $
$\therefore (a-1)^2+b^2=a^2+b^2 $
$\therefore a=1 / 2 $
$\text { since }| a + bi |=8 $
$a^2+b^2=64$
$b ^2=64-\frac{1}{4}=\frac{255}{4}=\frac{ u }{ v } $
$\therefore u =255 \text { and } v =4 $
$\therefore u + v =259$