Multiple choice

A variable chord is drawn through the origin to the circle $x^{2} + y^{2} - 2ax = 0$. The locus of the centre of the circle drawn on this chord as diameter is

  1. $x^{2} + y^{2} + ax = 0$
  2. $x^{2} + y^{2} + ay = 0$
  3. $x^{2} + y^{2} - ax = 0$
  4. $x^{2} + y^{2} - ay = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the chord be y = mx. The intersection with x^2 + y^2 - 2ax = 0 gives x^2 + m^2x^2 - 2ax = 0, so x = 2a/(1+m^2) and y = 2am/(1+m^2). The circle with this chord as diameter has center at (a/(1+m^2), am/(1+m^2)). Eliminating m leads to x^2 + y^2 - ax = 0.

AI explanation

Let the endpoints of the variable chord be the origin (0, 0) and a point P on the original circle. The center of the new circle drawn on this chord as diameter is the midpoint of OP, so if the center is (h, k), then P has coordinates (2h, 2k). Because P lies on the original circle x^2 + y^2 - 2ax = 0, we substitute the coordinates to get (2h)^2 + (2k)^2 - 2a(2h) = 0. Simplifying this gives 4h^2 + 4k^2 - 4ah = 0, and dividing by 4 yields the locus h^2 + k^2 - ah = 0, which is x^2 + y^2 - ax = 0.

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