I've been stuck on this question for the better part of a day now:
Let $a$ be a non-zero complex number such that $|a| \neq 1$. Let $P$ be the point $a$ in the complex plane, and let $Q$ be the point $\frac{1}{\bar a}$, Let $C_1$ be the circle $\{z:|z| = 1\}$ and let $C_2$ be any circle that passes through both $P$ and $Q$. Show that $C_1$ and $C_2$ intersect orthogonally.
I don't even know how to begin solving this question, but here are a few things which I've tried to do:
- Figure out that $\{z:|z| = 1\}$ refers to the unit circle.
- Tried to find an expression for the centre $z$ of $C_2$ using $|z-a| = |z-\frac{1}{\bar a}|$ to no avail — I opened up the expression as $(z-a)(\bar z-\bar a) = (z-\frac{1}{\bar a})(\bar z-\frac{1}{a})$. I also tried using the fact that $|z-a| + |z- \frac{1}{\bar a}| = |a-\frac{1}{\bar a}|.$
- Reasoned that if $z_1$ is the point of interception then $\tan (\frac{-1}{\arg{z_1}})$ would be the slope of the tangent of $C_2$.
Are any of these approaches right? I would appreciate a hint.
Edit: I should mention that I have only an advanced high-school understanding of complex numbers. This question is from a book that meant to be solved by high-schoolers.