Question

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

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

Answer 1

Answer: (B)

a circle

Choosing OA as x-axis $A=(r,0),B=(0,r)$ and any point P on the circle is $(r\cos \theta ,r\sin \theta )$
If $(x,y)$ is the centroid of $△PAB$, then
$3x=r\cos \theta +r+0$ and $3y=r\sin \theta +0+r$
$\therefore {(3x-1)}^2+{(3y-r)}^2=r^2$
Hence, locus of $P$ is a circle