Let A(1,4) and B(1,-5) be two points.


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 $(P A)^{2}+(P B)^{2}$ have maximum value, then the points, $\mathrm{P}, \mathrm{A}$ and $\mathrm{B}$ lie on :

  1. (1) a parabola

  2. (2) a straight line

  3. (3) a hyperbola

  4. (4) an ellipse

Correct Option: , 2


$\therefore \mathrm{PA}^{2}=\cos ^{2} \theta+(\sin \theta-3)^{2}=10-6 \sin \theta$

$\mathrm{PB}^{2}=\cos ^{2} \theta+(\sin \theta-6)^{2}=37-12 \sin \theta$

$\mathrm{PA}^{2}+\left.\mathrm{PB}^{2}\right|_{\max .}=47-\left.18 \sin \theta\right|_{\min } \Rightarrow \theta=\frac{3 \pi}{2}$

$\therefore P, A, B$ lie on a line $x=1$

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