The equation of the directrix of a hyperbola is x − y + 3 = 0. Its focus is (−1, 1) and eccentricity 3. Find the equation of the hyperbola.

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Solution

Let S be the focus and $P (x, y)$ be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM

$\Rightarrow \sqrt{{(x - (- 1))}^{2} + {(y - 1)}^{2}} = 3 \times (\frac{x - y + 3}{\sqrt{2}})$
Squaring both the sides, we get:
${(x + 1)}^{2} + {(y - 1)}^{2} = \frac{9}{2} {(x - y + 3)}^{2} \Rightarrow x^{2} + 2 x + 1 + y^{2} - 2 y + 1 = \frac{9}{2} (x^{2} + y^{2} + 9 - 2 x y - 6 y + 6 x) \Rightarrow 2 x^{2} + 4 x + 2 + 2 y^{2} - 4 y + 2 = 9 x^{2} + 9 y^{2} + 81 - 18 x y - 54 y + 54 x \Rightarrow 7 x^{2} + 7 y^{2} + 50 x - 50 y - 18 x y + 77 = 0$
Equation of the hyperbola:
$7 x^{2} + 7 y^{2} + 50 x - 50 y - 18 x y + 77 = 0$

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