Question

Let $AB$ be a chord of the circle $x^2+y^2=r^2$ subtending a right angle at the centre. Then, locus of the centroid of the triangle $PAB$ as $P$ moves on the circles is

• A. a parabola
• B. a circle
• C. an ellipse
• D. a pair of straight line

Answer 1

Answer: (B)

Let $A(r,0)$ and $B(0,r)$ be end points of chord $AB$ and moving point $P(r\cos \theta ,r\sin \theta )$ on the circle.
Let centroid is $(x,y)$

$\Rightarrow x=\frac{r+r\cos \theta +0}{3}$
$\Rightarrow (3x-r{)}^2=r^2{\cos }^2\theta ........$(1)
$\Rightarrow y=\frac{0+r\sin \theta +r}{3}$
$\Rightarrow (3y-r{)}^2=r^2{\sin }^2\theta ........$(2)
Equation (1) $+(2)$
$\Rightarrow (3x-r{)}^2+(3y-r{)}^2=r^2$ which is a circle