Find the locus of the midpoints of chords of hyperbola `3x^(2)-2y^(2)+4x-6y=0` parallel to y = 2x.

Options

A. `3x-4y=4`
B. `3y-4x+4=0`
C. `4x-4y=3`
D. `3x-4y=2`

Explanation

Let `(h,k)` be the mid-point of a chord of the hyperbola `3x^(2)-2y^(2)+4x-6y=0`. Then, the equation of the chord is
`3hx-2ly+2(x+h)-3(y+k)=3h^(2)-2k^(2)+4h-6k` [Using `T=S'`]
`implies(3h+2)x-(2k+3)y-3h^(2)+2k^(2)-2h+3k=0`
This is parallel to `y=2x`.
`:. (-(3h+2))/(-(2k+3))=2implies3h+2=4k+6implies3h-4k=4`
Hence, the locus of `(h,k) is `3x-4y=4`