Question

Find the equation of the hyperbola whose focus is (0, 3), directrix is x + y − 1 = 0 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

\[\Rightarrow\] \[\sqrt{(x - 0 )^2 + (y - 3 )^2} = 2\left( \frac{x + y - 1}{\sqrt{2}} \right)\]

Squaring both the sides:

\[(x - 0 )^2 + (y - 3 )^2 = 4 \left( \frac{x + y - 1}{\sqrt{2}} \right)^2 \]

\[ \Rightarrow x^2 + y^2 + 9 - 6y = 2\left( x^2 + y^2 + 1 + 2xy - 2y - 2x \right)\]

\[ \Rightarrow x^2 + y^2 + 4xy + 2y - 4x - 7 = 0\]

∴ Equation of the hyperbola = \[x^2 + y^2 + 4xy + 2y - 4x - 7 = 0\]