Q: The locus of the middle points of chords of hyperbola 3x^{2} – 2y^{2} + 4x – 6y = 0 parallel to y = 2x is

(A) 3x – 4y = 4

(B) 3y – 4x + 4 = 0

(C) 4x – 4y = 3

(D) 3x – 4y = 2

Ans: (A)

Sol. Let mid point be (h, k).

Equation of a chord whose mid point is (h, k) would be T = S_{1}

or 3xh – 2yk + 2( x+h) – 3( y+k) = 3h^{2} – 2k^{2} + 4h – 6k

⇒ x(3k + 2) – y ( 2k+3) – 2h + 3k – 3h^{2} + 2k^{2} = 0

it’s slope is $\large \frac{3h+2}{2k+3} = 2 $(given)

⇒ 3h = 4k + 4

⇒ Required locus is 3x – 4y = 4.

Hence (A) is the correct answer.