Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e) = 3

To find: equation of the hyperbola

Let M be the point on directrix and P(x, y) be any point of the hyperbola

Formula used:

where e is an eccentricity, PM is perpendicular from any point P on hyperbola to the directrix

Therefore,

Squaring both sides:

{∵ (a – b)² = a² + b² + 2ab &

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac}

⇒ 2{x² + 1 + 2x + y² + 1 – 2y} = 9{x² + y²+ 9 + 6x – 6y – 2xy}

⇒ 2x² + 2 + 4x + 2y² + 2 – 4y = 9x² + 9y²+ 81 + 54x – 54y – 18xy

⇒ 2x² + 4 + 4x + 2y²– 4y – 9x² - 9y² - 81 – 54x + 54y + 18xy = 0

⇒ – 7x² - 7y² – 50x + 50y + 18xy – 77 = 0

⇒ 7x² + 7y² + 50x – 50y – 18xy + 77 = 0

This is the required equation of hyperbola