Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in following cases :
Conjugate axis is $7$ and passes through the point $(3,-2)$

Answer 1

Let equation of hyperbola when axes as axes of coordinate

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\dots \left(1\right)$

$\because$ Conjugate axis $=7$

$\Rightarrow 2b=7$

$\Rightarrow b=\frac{7}{2}$

$\Rightarrow b^2=\frac{49}{4}$

Since hyperbola passes through point $(3,-2)$

$\Rightarrow \frac{3^2}{a^2}-\frac{(-2{)}^2}{\frac{49}{4}}=1$

$\Rightarrow \frac{3^2}{a^2}-\frac{4\times 4}{49}=1$

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

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

$\Rightarrow a^2=\frac{441}{65}$

Put value of $a^2$ and $b^2$ in equation $\left(1\right)$

$\Rightarrow \frac{x^2}{\frac{441}{65}}-\frac{y^2}{\frac{49}{4}}=1$

$\Rightarrow \frac{65x^2}{441}-\frac{4y^2}{49}=1$

$\Rightarrow 65x^2-36y^2=441$

∴ Equation of hyperbola is $65x^2-36y^2=441$