Question

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

• A. a parabola
• B. a straight line
• C. a hyperbola
• D. an ellipse

Answer 1

The correct option is B a straight line


$\therefore P{A}^2={\cos }^2\theta +(\sin \theta -3{)}^2=10-6\sin \theta$
$P{B}^2={\cos }^2\theta +(\sin \theta +6{)}^2=37+12\sin \theta$
$P{A}^2+P{B}^2{|}_{max}=47+6\sin \theta {|}_{max}\Rightarrow \theta =\frac{\pi }{2}$
$\therefore P,A,B$ lie on the line $x=1$.