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

To find: equation of conic

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}

⇒ x² + 1 – 2x + y² + 1 + 2y = x² + y² + 1 – 2xy + 2x – 2y

⇒ x² + 1 – 2x + y² + 1 + 2y – x² – y² + 2xy – 1 – 2x + 2y = 0

⇒ 2xy – 4x + 4y + 1 = 0

This is the required equation of hyperbola