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

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — A circle

Detailed Solution

Take the centre at the origin and fix the chord as $A = (r, 0)$ , $B = (0, r)$ (any such chord can be rotated to this position). Let $P = (r cos θ, r s in θ)$ .

The centroid is $(h, k) = (\frac{r + r cos θ}{3}, \frac{r + r s in θ}{3})$ , so $(3 h - r)^{2} + (3 k - r)^{2} = r^{2}$ .

This is a circle with centre $(\frac{r}{3}, \frac{r}{3})$ and radius $\frac{r}{3}$ .

✓ Check

For

r = 3

and

P = (3, 0)

G = (\frac{3 + 0 + 3}{3}, \frac{0 + 3 + 0}{3}) = (2, 1)

, whose distance from the centre

(1, 1)

1 = \frac{r}{3}

⚠ Common trap

Option (C), an ellipse, would need different scaling in the two coordinates; the centroid divides both coordinates of

P

by 3, a uniform scaling, so a circle stays a circle.

Quick tip

$G = \frac{A + B + P}{3}$ : centre $\frac{A + B}{3}$ , radius $\frac{r}{3}$

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Circle — Preparation

This question tests your understanding of Circle, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

In coordinate geometry, a circle is the set of points at a fixed distance (the radius) from a fixed point (the centre). This topic covers the standard equation (x - h)^2 + (y - k)^2 = r^2, the general equation x^2 + y^2 + 2gx + 2fy + c = 0 with centre (-g, -f), the diameter form, and parametric equations. Exam questions also test the position of a point relative to a circle, the condition for a line to be a tangent, equations of tangents and normals, the length of a tangent from an external point, and the intersection of two circles. The fastest approach is often to compare the distance from the centre to a line with the radius: equal means tangent, smaller means the line cuts the circle. Practise the circle questions below, each with a step-by-step solution.

‹ All Circle questions Browse all topics ›

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

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

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