Multiple choice

Let $\mathrm{A}\mathrm{B}$ 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 $P\mathrm{A}\mathrm{B}$ as $\mathrm{P}$ moves on the circle is

  1. A parabola

  2. A circle

  3. An ellipse

  4. A pair of straight line
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let A=(r,0) and B=(0,r). Centroid G = ((r+x_p)/3, (r+y_p)/3). Since x_p^2 + y_p^2 = r^2, substituting x_p = 3x - r and y_p = 3y - r gives (3x-r)^2 + (3y-r)^2 = r^2, which is the equation of a circle.

AI explanation

Let the coordinates of P be (r*cos(a), r*sin(a)). Because chord AB subtends a right angle at the center (0, 0), the endpoints of the chord can be taken as (r, 0) and (0, r). The centroid G of triangle PAB has coordinates ((r*cos(a) + r + 0)/3, (r*sin(a) + 0 + r)/3), which simplifies to (r*(cos(a) + 1)/3, r*(sin(a) + 1)/3). If we shift the origin to the point (r/3, r/3) by letting X = x - r/3 and Y = y - r/3, the coordinates of G become X = r*cos(a)/3 and Y = r*sin(a)/3. Squaring and adding these gives X^2 + Y^2 = r^2/9, which represents a circle.

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