First slide

Definition of a circle

Question

Let $A B$ 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 triangle $P A B$ 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 ( $h, k$ ) be the centroid, then
$h = \frac{r + r \cos θ}{3}, k = \frac{r + r \sin θ}{3} .$
$\begin{array}{l} \Rightarrow (3 h - r)^{2} + (3 k - r)^{2} = r^{2} \\ \Rightarrow 9 (h^{2} + k^{2}) - 6 r (h + k) + r^{2} = 0 \end{array}$
Locus of $(h, k)$ is a circle.

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