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

Question

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, two lines making an angle π 3 at origin, a conic with eccentricity a b, two perpendicular lines, two coincident lines

MARKS App · Accepted Answer

For parabola, P S = P N ⇒ x - 2 2 + y + 3 2 = 3 x - 2 y + 5 3 2 + - 2 2 ⇒ 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 ⇒ 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 ⇒ 2 x - 3 y 2 = 0 It represents two coincident lines.

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