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`

Answer

A

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
`3h-2ky+2(x+h)-3(y+k)`
`3h^(2)-2k^(2)+4h-6k ["using" T=S']`
`implies (3h+2)x-(2k+3)y-3h^(2)+3h^(2)+2k^(2)-2h+3k=0`
This is parallel to `y=2x,`
`therefore (-(3h+2))/-(2k+3)=2`
`implies3h+2=4k+6 implies3h-4k=4`
Hence, locus of (h,k) is `3x-4y=4`