Multiple choice

A circle is drawn on the chord of a circle x2 + y2 = a2 as diameter. The chord lies on the line x + y = a. What is the equation of the circle?

  1. x2 + y2 - ax - ay + a2 = 0

  2. x2 + y2 - ax - ay = 0

  3. x2 + y2 + ax + ay = 0

  4. x2 + y2 + ax + ay - 2a2 = 0
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B Correct answer
Explanation

The chord of x^2 + y^2 = a^2 is x + y = a. The points of intersection are (a,0) and (0,a). A circle with this chord as diameter has the equation (x-a)(x-0) + (y-0)(y-a) = 0 => x^2 - ax + y^2 - ay = 0 => x^2 + y^2 - ax - ay = 0.

AI explanation

The endpoints of the chord on the circle x2 + y2 = a2 that lies on the line x + y = a are found by substituting y = a - x. This gives x2 + (a - x)2 = a2, which simplifies to 2x2 - 2ax = 0. The solutions for x are 0 and a, giving the endpoints (0, a) and (a, 0). The equation of a circle with diameter endpoints (x1, y1) and (x2, y2) is (x - x1)(x - x2) + (y - y1)(y - y2) = 0. Substituting the points (0, a) and (a, 0) gives (x - 0)(x - a) + (y - a)(y - 0) = 0, which expands to x2 + y2 - ax - ay = 0.

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