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 [2001]

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 [2001]

Smart Achievers · Accepted Answer

(2)

$Given a circle x^{2} + y^{2} = r^{2} with centre at (0, 0) and radius r .$

$Let A and B be (- r, 0) and (0, - r), so that ∠ A O B = 90 °$

$and an arbitrary point P on the given circle be (r \cos θ, r \sin θ) .$

$For locus of centroid of ∆ A B P$

$(\frac{r \cos θ - r}{3}, \frac{r \sin θ - r}{3}) = (x, y)$

$\Rightarrow r \cos θ - r = 3 x, r \sin θ - r = 3 y$

$\Rightarrow r \cos θ = 3 x + r, r \sin θ = 3 y + r$

$On squaring and adding,$

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

Solution

Q.
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 [2001]

1 a parabola

2 a circle

3 an ellipse

4 a pair of straight lines

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