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

Answer 1

Answer: (A)

$P$ be a point on $(x-1{)}^2+(y-1{)}^2=1$
so $P(1+\cos \theta ,1+\sin \theta )$
$\begin{array}{ll}A(1,4)& B(1,-5)\end{array}$
$(PA{)}^2+(PB{)}^2$
$=(\cos \theta {)}^2+(\sin \theta -3{)}^2+(cos\theta {)}^2+(\sin \theta +6{)}^2$
$=47+6\sin \theta$
is maximum if $\sin \theta =1$
$\Rightarrow \sin \theta =1,\cos \theta =0$
$P(1,1)A(1,4)B(1,-5)$
$P,A,B$ are collinear points.