The equation of the hyperbola whose directrix is $2 x + y = 1$ ,corresponding focus is $(1, 1)$ and eccentricity $\sqrt{3}$ , is given by

7 x^{2} + 12 x y - 2 y^{2} - 2 x + 4 y - 7 = 0

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2 x^{2} + 12 x y - 7 y^{2} - 2 x + 14 y - 7 = 0

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7 x^{2} - 12 x y + 2 y^{2} - 2 x + 14 y - 22 = 0

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7 x^{2} + 12 x y - 2 y^{2} - 2 x - 14 y - 22 = 0

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Solution

The correct option is D $7 x^{2} + 12 x y - 2 y^{2} - 2 x + 4 y - 7 = 0$
Let $P (x, y)$ is any point on the hyperbola.
given, focus of parabola is $S (1, 1)$ .
equation of directrix is $2 x + y = 1$
From P draw PM perpendicular to the directrix then $P M = (2 x + y - 1) / \sqrt (2 ² + 1 ²) = (2 x + y - 1) / \sqrt 5$
Also from the definition of the hyperbola, we have
$S P / P M = e \Rightarrow S P = e P M$
$\Rightarrow \sqrt (x - 1) ² + (y - 1) ² = \sqrt 3 (2 x + y - 1) / \sqrt 5$
$\Rightarrow (x - 1) ² + (y - 1) ² = 3 (2 x + y - 1) ² / 5$
$\Rightarrow 5 [(x ² - 2 x + 1) + (y ² - 2 y + 1)] = 3 (4 x ² + y ² + 1 + 4 x y - 4 x - 2 y)$
$\Rightarrow 5 x ² - 10 x + 5 + 5 y ² - 10 y + 5 = 12 x ² + 3 y ² + 3 + 12 x y - 12 x - 6 y$
$\Rightarrow 7 x ² + 2 y ² + 12 x y - 2 x + 4 y - 7 = 0$
hence, equation of hyperbola is $7 x ² - 2 y ² + 12 x y - 2 x + 4 y - 7 = 0$

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7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

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7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

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