Question 53
Let $a,b\in\mathbb R$ and $z\in\mathbb C\setminus\mathbb R$ satisfy $|a-b|=|a+b-2z|$.
a) Prove that $|z-a|^x+|\bar z-b|^x=|a-b|^x$, with unknown $x\in\mathbb R$, has a unique solution.
b) Solve $|z-a|^x+|\bar z-b|^x\le|a-b|^x$ for $x\in\mathbb R$.
a) Prove that $|z-a|^x+|\bar z-b|^x=|a-b|^x$, with unknown $x\in\mathbb R$, has a unique solution.
b) Solve $|z-a|^x+|\bar z-b|^x\le|a-b|^x$ for $x\in\mathbb R$.