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The locus of the mid-points of the chords of the hyperbola 3 x 2 - 2 y 2 + 4 x - 6 y = 0 which are parallel to the line y = 2 x + 4 is

Options

  1. A3 x - 4 y = 4
  2. B4 x - 4 y = 3
  3. C3 y - 4 x + 4 = 0
  4. D3 x - 4 y = 2

Correct answer

A. 3 x - 4 y = 4

Step-by-step solution

Let mid-point of the chord is h , k ⇒ equation of the chord is 3 h x - 2 k y + 4 x + h 2 - 6 y + k 2 = 3 h 2 - 2 k 2 + 4 h - 6 k ⇒ 3 h + 2 x + - 2 k - 3 y = 3 h 2 - 2 k 2 + 2 h - 3 k ⇒ Slope of the chord is equal to 3 h + 2 2 k + 3 = 2 ⇒ locus of the mid-points is 3 x + 2 = 4 y + 6 ⇒ 3 x - 4 y - 4 = 0

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