Question

Find the equation of the hyperbola whose focus is (a, 0), directrix is 2x − y + a = 0 and eccentricity = \[\frac{4}{3}\].

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

\[\Rightarrow\] \[\sqrt{(x - a )^2 + (y - 0 )^2} = \frac{4}{3}\left( \frac{2x - y + a}{\sqrt{5}} \right)\]

Squaring both the sides:

\[(x - a )^2 + (y )^2 = \frac{16}{9} \left( \frac{2x - y + a}{5} \right)^2 \]

\[ \Rightarrow x^2 - 2ax + a^2 + y^2 = \frac{16}{45}\left( 4 x^2 + y^2 + a^2 - 4xy - 2ya + 4xa \right)\]

\[ \Rightarrow 45 x^2 - 90ax + 45 a^2 + 45 y^2 = 64 x^2 + 16 y^2 + 16 a^2 - 64xy - 32ay + 64ax\]

\[ \Rightarrow 19 x^2 - 29 y^2 - 64xy - 32ay + 154ax - 29 a^2 = 0\]

∴ Equation of the hyperbola = \[19 x^2 - 29 y^2 - 64xy - 32ay + 154ax - 29 a^2 = 0\]