Let A (1,4)
Question:

Let $\mathrm{A}(1,4)$ and $\mathrm{B}(1,-5)$ be two points. Let $\mathrm{P}$ be a point on the circle $(x-1)^{2}+(y-1)^{2}=1$ such that $(\mathrm{PA})^{2}+(\mathrm{PB})^{2}$ have maximum value, then the points $P, A$ and $B$ lie on :

1. a straight line

2. a hyperbola

3. an ellipse

4. a parabola

Correct Option: 1

Solution:

$P$ be a point on $(x-1)^{2}+(y-1)^{2}=1$

so $\mathrm{P}(1+\cos \theta, 1+\sin \theta)$

$\mathrm{A}(1,4) \quad \mathrm{B}(1,-5)$

$(\mathrm{PA})^{2}+(\mathrm{PB})^{2}$

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

$=47+6 \sin \theta$

is maximum if $\sin \theta=1$

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

$\mathrm{P}(1,1) \mathrm{A}(1,4) \mathrm{B}(1,-5)$

$\mathrm{P}, \mathrm{A}, \mathrm{B}$ are collinear points.