Multiple choice

The locus of the middle point of the chords of the circle x2 + y2 = a2 such that the chords pass through a given point (x1, y1) is

  1. x2 + y2 - xx1 - yy1 = 0

  2. x2 + y2 = x12 + y12

  3. x + y = x1 + y1

  4. x + y = x12 + y12

  5. x2 - y2 - x12 - y12 = 0
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A Correct answer
Explanation

The chord with midpoint (h,k) has equation xh + yk = h^2 + k^2. Since it passes through (x1, y1), x1h + y1k = h^2 + k^2. Replacing (h,k) with (x,y) gives x^2 + y^2 - xx1 - yy1 = 0.

AI explanation

Let (h, k) be the midpoint of a chord of the circle x squared plus y squared equals a squared. Since the midpoint lies inside the circle, we can use the property of a circle and its chord to establish the locus by squaring the distance components. Using the standard equation of the diameter with one end as the center (0, 0) and the other as (x1, y1), we replace the variables with constants to get x squared plus y squared minus xx1 minus yy1 equals 0.

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