Question

If a chord $AB$ subtends a right angle at the center of a given circle, then the locus of the centroid of the triangle $PAB$ as $P$ moves on the circle is a/an

• A. parabola
• B. ellipse
• C. hyperbola
• D. circle

Answer 1

The correct option is D circle

We choose the centre $O$ of the circle as the origin and the lines $OA,OB$ as the X-axis and the Y-axis, respectively.

If $a$ be a radius of the given circle, then

$A\equiv (a,0)$ and $B=(0,a)$

and a variable point on the circle

$P\equiv (a\cos \theta ,a\sin \theta )$

If $(h.k)$ be the coordinates of the centroid of triangle $PAB,$ then we have

$3h=a(1+\cos \theta )$ ...(1)

and, $3k=a(1+\sin \theta )$ ...(2)

Eliminating $\theta$ from equations (1) and (2), we have

${(3h-a)}^2+{(3k-a)}^2=a^2.$

Putting $(x,y)$ in place of $(h,k)$ given the equation of the required locus as

${(3x-a)}^2+{(3y-a)}^2=a^2,$ which is a circle.