Multiple choice

The locus of the middle point of the chord of the circle ${x}^{2}+{y}^{2}-12x+4=0$, that subtends an angle $\frac{2\pi}{3}$ at the centre is

  1. ${x}^{2}+{y}^{2}-12x+4y+30=0$
  2. ${x}^{2}+{y}^{2}-12x+4y+31=0$
  3. ${x}^{2}+{y}^{2}+12x+4y+30=0$
  4. ${x}^{2}+{y}^{2}-12x+28=0$
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D Correct answer
Explanation

The chord subtends 120 degrees at the center (6,0). The midpoint of the chord forms a right triangle with the center and the chord endpoint. The distance from center to midpoint is r * cos(60) = sqrt(32) * 1/2 = sqrt(8). The locus of the midpoint is a circle centered at (6,0) with radius sqrt(8). (x-6)^2 + y^2 = 8 -> x^2 - 12x + 36 + y^2 = 8 -> x^2 + y^2 - 12x + 28 = 0.

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