Q97 / 160 · TS EAMCET 2021 (05 Aug Shift 1) · Mathematics

If $a x^{2} + 2 h x y + b y^{2} - 82 x + 98 y + 144 = 0$ is the equation of a parabola with focus $(2, - 3)$ and directrix $3 x - 2 y + 5 = 0,$ then $a x^{2} + 2 h x y + b y^{2} = 0$ represents

Solution

For parabola, $P S = P N$

$\Rightarrow \sqrt{{(x - 2)}^{2} + {(y + 3)}^{2}} = |\frac{3 x - 2 y + 5}{\sqrt{{(3)}^{2} + {(- 2)}^{2}}}|$

$\Rightarrow 13 [(x^{2} - 4 x + 4) + (y^{2} + 6 y + 9)] = 9 x^{2} + 4 y^{2} + 25 - 12 x y - 20 y + 30 x$

$\Rightarrow 4 x^{2} + 9 y^{2} + 12 x y - 82 x + 98 y + 144 = 0$

Comparing with given parabola

$a = 4$ , $b = 9$ , $h = - 6$

Now, $4 x^{2} - 12 x y + 9 y^{2} = 0$

$\Rightarrow {(2 x - 3 y)}^{2} = 0$

It represents two coincident lines.

Hence, the correct answer is option: 'D'.