Explanation

(a)
Let $(h,k)$ is mid point of chord.
Then, its equation is
$3hx - 2ky + 2(x + h) - 3(y + k) = 3h^{2} - 2k^{2} + 4h - 6k$
$\Rightarrow x(3h + 2) + y( - 2k - 3) = 3h^{2} - 2k^{2} + 2h - 3k$
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 mid point is $3x - 4y = 4.$