Given: Equation of directrix of a hyperbola is 2x + y – 1 = 0. Focus of hyperbola is (1, 1) and eccentricity (e) =

To find: equation of hyperbola

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

Formula used:

where e is 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}

⇒ 5{x² + 1 – 2x + y² + 1 – 2y} = 3{4x² + y²+ 1 + 4xy – 2y – 4x}

⇒ 5x² + 5 – 10x + 5y² + 5 – 10y = 12x² + 3y² + 3 + 12xy – 6y – 12x

⇒ 5x² + 5 – 10x + 5y² + 5 – 10y – 12x² – 3y² – 3 – 12xy + 6y + 12x = 0

⇒ – 7x² + 2y² + 2x – 4y – 12xy + 7 = 0

⇒ 7x² – 2y² – 2x + 4y + 12xy – 7 = 0

This is the required equation of hyperbola.