Multiple choice

The locus of the midpoints of chords of the circle $x^2+y^2 = 1$ which subtends a right angle at the origin is

  1. $x^2+y^2 = \dfrac{1}{4}$
  2. $x^2+y^2 = \dfrac{1}{2}$
  3. $xy = 0$
  4. $x^2-y^2 = 0$
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B Correct answer
Explanation

For a chord to subtend a right angle at the origin, the distance from the origin to the chord must be 1/sqrt(2). The locus of the midpoint (h, k) of such chords is x^2 + y^2 = d^2, where d is the distance from the origin. Thus, x^2 + y^2 = (1/sqrt(2))^2 = 1/2.

AI explanation

For a chord in the circle x squared plus y squared equals 1 to subtend a right angle at the origin, the perpendicular distance d from the origin to the chord must be 1 divided by the square root of 2. If a point (h, k) is the midpoint of such a chord, its distance from the origin is exactly this perpendicular distance, so h squared plus k squared equals 1 half. Replacing (h, k) with variables (x, y) gives the equation x squared plus y squared equals 1 half.

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