Question

The locus of middle point of chords of hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ parallel to $y = 2x$ is

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

Answer 1

Answer: (A)

Let $(h, k)$ be mid-point of chord.

Then, its equation is $T = S_1$

$\therefore 3hx-2ky+2\left(x+h\right)-3\left(y+k\right)$

$=3h^{2}-2k^{2}+4h-6k$

$\Rightarrow x\left(3h+2\right)+y\left(-2k-3\right)$

$=3h^{2}-2k^{2}+2h-3k$

Since, this line is parallel to $y = 2x$

$\therefore \frac{3h+2}{2k+3}=2$

$\Rightarrow 3h + 2 = 4k + 6$

$\Rightarrow 3h-4k=4$

Thus, locus of point is
$3k - 4y = 4$