Question

Find the equation of the hyperbola whose foci at (± 2, 0) and eccentricity is 3/2. 

Answer 1

 The foci of the hyperbola are \[\left( \pm 2, 0 \right)\]. 

∴ \[ae = 2\]

\[ \Rightarrow a = 2 \times \frac{2}{3} = \frac{4}{3}\]

\[ \Rightarrow a^2 = \frac{16}{9}\]

Now,

\[\left( ae \right)^2 = a^2 + b^2 \]

\[ \Rightarrow \left( 2 \right)^2 = \left( \frac{4}{3} \right)^2 + b^2 \]

\[ \Rightarrow 4 - \frac{16}{9} = b^2 \]

\[ \Rightarrow b^2 = \frac{20}{9}\]

Therefore, the equation of the hyperbola is given by

\[\frac{9 x^2}{16} - \frac{9 y^2}{20} = 1\]

\[ \Rightarrow \frac{x^2}{4} - \frac{y^2}{5} = \frac{4}{9}\]