Question 65
Let $a,b\in\mathbb R$ and $z\in\mathbb C\setminus\mathbb R$ satisfy $|a-b|=|a+b-2z|$.
a) Prove that $|z-a|^x+|\bar z-b|^x=|a-b|^x$ has exactly one real solution $x$.
b) Determine all real $x$ such that $|z-a|^x+|\bar z-b|^x\le |a-b|^x$.
a) Prove that $|z-a|^x+|\bar z-b|^x=|a-b|^x$ has exactly one real solution $x$.
b) Determine all real $x$ such that $|z-a|^x+|\bar z-b|^x\le |a-b|^x$.