Equation of the hyperbola with one focus at the origin, directrix $x + 3 = 0$ and eccentricity $\sqrt{3}$ is

x^{2} - 2 y^{2} + 18 x + 27 = 0

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x^{2} - y^{2} + 18 x + 27 = 0

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x^{2} - 2 y^{2} - 18 x - 27 = 0

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2 x^{2} - y^{2} + 18 x + 27 = 0

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Solution

The correct option is D $2 x^{2} - y^{2} + 18 x + 27 = 0$

$l e t a n y p o i n t o n t h e h y p e r b o l a b e (x, y) W e k n o w, D i s t a n c e b e t w e e n t h e a n y p o i n t o n t h e h y p e r b o l a a n d t h e f o c u s i s e q u a l t o e c c e n t r i c i t y t i m e s t h e d i s t a n c e b e t w e e n t h e p o i n t a n d d i r c t r i x . O r, S 1 P = e (P N) O r, \sqrt{x^{2} + y^{2}} = \sqrt{3} (\frac{x + 3}{\sqrt{1}}) S q u a r i n g b o t h s i d e s w e g e t : x^{2} + y^{2} = 3 (x^{2} + 9 + 6 x) O r, 2 x^{2} - y^{2} + 18 x + 27 = 0$

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x^{2} = 4 y

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x^{2} = 4 y

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