Let A 1 , 4 and B 1 , - 5 be two points. Let P be a point on the circle x - 1 2 + y - 1 2 = 1 , such that P…

$P$ be a point on ${\left(x-1\right)}^2+{\left(y-1\right)}^2=1$, so $P\left(1+\cos \theta ,1+\sin \theta \right)$.

Given $A\left(1,4\right) \& B\left(1,-5\right)$

${\left(PA\right)}^2+{\left(PB\right)}^2$

$={\left(\cos \theta \right)}^2+{\left(\sin \theta -3\right)}^2+{\left(\cos \theta \right)}^2+{\left(\sin \theta +6\right)}^2$

$=47+6\sin \theta$ is maximum if $\sin \theta =1$

$\Rightarrow \sin \theta =1,\cos \theta =0$

$P\left(1,2\right), A\left(1,4\right), B\left(1,-5\right)$

 The points $P, A$ and $B$ lie on the line $x=1$.