Q. 2 F5.0( 1 Vote )

# Find the equation of the hyperbola whose

focus is (2, 2), directrix is x + y = 9 and eccentricity = 2

Answer :

**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:**

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

Therefore,

Squaring both sides:

{∵ (a – b)^{2} = a^{2} + b^{2} + 2ab &

(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac}

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

⇒ x^{2} – 4x + y^{2} + 8 – 4y = 2x^{2} + 2y^{2} + 162 + 4xy – 36y – 36x

⇒ x^{2} – 4x + y^{2} + 8 – 4y – 2x^{2} – 2y^{2} – 162 – 4xy + 36y + 36x = 0

⇒ – x^{2} – y^{2} + 32x + 32y + 4xy – 154 = 0

**⇒** **x ^{2} + y^{2} – 32x – 32y + 4xy + 154 = 0**

This is the required equation of hyperbola.

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