Question

Find the equation of the hyperbola whose focus is (2, 2), directrix is x + y = 9 and eccentricity = 2

Answer 1

Given: Equation of directrix of a hyperbola is x + y – 9 = 0. 

Focus of hyperbola is (2, 2) 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:

e = \(\frac{PF}{PM}\) \(\Rightarrow\) PF = e PM

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

Therefore,

{∵ (a – b)² = a² + b² + 2ab & 

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

⇒ x² + 4 – 4x + y² + 4 – 4y = 2{x² + y² + 81 + 2xy – 18y – 18x} 

⇒ x² – 4x + y² + 8 – 4y = 2x² + 2y² + 162 + 4xy – 36y – 36x 

⇒ x² – 4x + y² + 8 – 4y – 2x² – 2y² – 162 – 4xy + 36y + 36x = 0 

⇒ – x² – y ² + 32x + 32y + 4xy – 154 = 0 

⇒ x² + y² – 32x – 32y + 4xy + 154 = 0 

This is the required equation of hyperbola.