Question

Let AB be a chord of the circle x² + y² = r² subtending a right angle at the centre, then the locus of the centroid of the ΔPAB as P moves on the circle is ______.

Options

• a parabola

• a circle

• an ellipse

• a pair of straight lines

Answer 1

Let AB be a chord of the circle x² + y² = r² subtending a right angle at the centre, then the locus of the centroid of the ΔPAB as P moves on the circle is a circle.

Explanation:

Let OA as X-axis, A = (r, 0), B = (0, r) and any point P on the circle is (r cos θ, r sin θ).

If (x, y) is the centroid of ΔPAB, then

3x = r cos θ + r + 0  ...(i)

and 3y = r sin θ + 0 + r  ...(ii)

From Equations (i) and (ii),

∴  (3x – r)² + (3y – r)² = r²

 Hence, locus of P is a circle.