Let A(1, 4) and B(1, - 5) be two points. Let P be a point on the...

$P$ 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 $A\left(1, 4\right) \text{ and } B\left(1, - 5\right)$

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

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

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

$⇒ sin\theta = 1, cos\theta = 0$

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

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