Given: Equation of directrix of a hyperbola is 2x + 3y – 1 = 0. Focus of hyperbola is (2, -1) and eccentricity (e) = 2

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}

⇒ 13{x² + 4 – 4x + y² + 1 + 2y} = 4{4x² + 9y² + 1 + 12xy – 6y – 4x}

⇒ 13x² + 52 – 52x + 13y² + 13 + 26y = 16x² + 36y² + 4 + 48xy – 24y – 16x

⇒ 13x² + 52 – 52x + 13y² + 13 + 26y – 16x² – 36y² – 4 – 48xy + 24y + 16x = 0

⇒ – 3x² – 23y² – 36x + 50y – 48xy + 61 = 0

⇒ 3x² + 23y² + 36x – 50y + 48xy – 61 = 0

This is the required equation of hyperbola.