First slide

Definition of a circle

Question

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

Moderate

A

a parabola
B

a circle
C

an ellipse
D

a pair of straight lines

Solution

$Let us take A \equiv (r, 0) and B (0, r) subtending \frac{π}{2} angle at the centre$
$Let P \equiv (r \cos θ, rsin θ)$
$So, G = (\frac{r (1 + \cos θ)}{3}, \frac{r (1 + \sin θ)}{3}) = (h, k)$
$Step-IV: So, {(\frac{3 h}{r} - 1)}^{2} + {(\frac{3 k}{r} - 1)}^{2} = 1 which is a circle$

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