Multiple choice

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

  1. a parabola

  2. an ellipse

  3. a circle

  4. a pair of straight lines
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let A = (r, 0) and B = (0, r) on the circle x^2 + y^2 = r^2. Let P = (r cos t, r sin t). The centroid G = ((r + 0 + r cos t)/3, (0 + r + r sin t)/3). This simplifies to (x - r/3)^2 + (y - r/3)^2 = (r/3)^2, which is the equation of a circle.

AI explanation

Let the coordinates of points A and B on the circle x squared plus y squared equals r squared be (r, 0) and (0, r) respectively, as the chord subtends a right angle at the center. For a variable point P with coordinates (a, b) on the circle, the centroid coordinates are calculated by averaging the vertices: x equals a divided by 3 plus r divided by 3, and y equals b divided by 3 plus r divided by 3. This means a equals 3x minus r and b equals 3y minus r, and substituting these into the circle equation yields 9 times x minus r divided by 3 squared plus 9 times y minus r divided by 3 squared equals r squared. This resulting equation represents a circle.

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