Question

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

Answer 1

Let S be the focus and  \[P\left( x, y \right)\] be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM

\[\Rightarrow\] \[\sqrt{(x - 2 )^2 + (y - 2 )^2} = 2\left( \frac{x + y - 9}{\sqrt{2}} \right)\]

Squaring both the sides:

\[(x - 2 )^2 + (y - 2 )^2 = 4 \left( \frac{x + y - 9}{2} \right)^2 \]

\[ \Rightarrow x^2 - 4x + 4 + y^2 - 4y + 4 = 2\left( x^2 + y^2 + 81 + 2xy - 18y - 18x \right)\]

\[ \Rightarrow x^2 - 4x + 4 + y^2 - 4y + 4 = 2 x^2 + 2 y^2 + 162 + 4xy - 36y - 36x\]

\[ \Rightarrow x^2 + y^2 + 4xy - 32y - 32x + 154 = 0\]

∴ Equation of the hyperbola = \[x^2 + y^2 + 4xy - 32y - 32x + 154 = 0\]