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AP EAMCET202120 Aug 2021Morning ShiftMathematicsCircleActual

Let C be the circle with centre 0 , 0 and radius 3 unit . The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2 π 3 at its centre, is:

Options

  1. Ax 2 + y 2 = 1
  2. Bx 2 + y 2 = 27 4
  3. Cx 2 + y 2 = 9 4
  4. Dx 2 + y 2 = 3 2

Correct answer

C. x 2 + y 2 = 9 4

Step-by-step solution

Let the coordinates of a point P be ( h ,   k ) which is mid point of the chord A B . Now, O P = h - 0 2 + k - 0 2 = h 2 + k 2 In ∆ A O P ,   cos ⁡ π 3 = O P O A ⇒     1 2 = h 2 + k 2 3 ⇒       h 2 + k 2 = 9 4 Hence, the required locus is x 2 + y 2 = 9 4 .

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