If x ^ 2 - y ^ 2 + 2hxy + 2gx + 2fy + c = 0 is the locus of a...

Let $P ( x, y )$ be the point whose distance from $x + 2y + 7 = 0$ and $2x - y + 8 = 0$ is equal.

$⇒ \dfrac{x + 2y + 7}{\sqrt{5}} = \pm \dfrac{2x - y + 8}{\sqrt{5}}$

$⇒ {x}^{2} + 4{y}^{2} + 49 + 4xy + 28y + 14x = 4{x}^{2} + {y}^{2} + 64 - 4xy - 16y + 32x$

$⇒ - 3{x}^{2} + 3{y}^{2} - 15 + 8xy + 44y - 18x = 0$

$⇒ 3{x}^{2} - 3{y}^{2} - 8xy + 18x - 44y + 15 = 0$

$⇒ {x}^{2} - {y}^{2} - \dfrac{8}{3}xy + 6x - \dfrac{44}{3}y + 5 = 0$

Now, on comparing ${x}^{2} - {y}^{2} + 2hxy + 2gx + 2fy + c = 0$ with above equation we get,

$⇒ h = \dfrac{- 4}{3}, g = 3, f = - \dfrac{22}{3}, c = 5$

$⇒ g + c + h - f = 3 + 5 - \dfrac{4}{3} + \dfrac{22}{3}$

$⇒ g + c + h - f = 14$