Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in  the conjugate axis is 7 and passes through the point (3, −2).

Answer 1

 Length of the conjugate axis, \[2b = 7\]

\[\Rightarrow b = \frac{7}{2}\]

Let the equation of the hyperbola be \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\].

It passes through  \[\left( 3, - 2 \right)\] .

\[\therefore \frac{3^2}{a^2} - \frac{( - 2 )^2}{\left( \frac{7}{2} \right)^2} = 1\]

\[ \Rightarrow \frac{3^2}{a^2} - \frac{16}{49} = 1\]

\[ \Rightarrow \frac{9}{a^2} = \frac{16}{49} + 1\]

\[ \Rightarrow \frac{9}{a^2} = \frac{65}{49}\]

\[ \Rightarrow a^2 = \frac{441}{65}\]

Therefore, the standard form of the hyperbola is \[\frac{65 x^2}{441} - \frac{4 y^2}{49} = 1\].

or \[ 65 x^2 - 36 y^2 = 441\]