Question

Write the eccentricity of the hyperbola 9x² − 16y² = 144.

Answer 1

Equation of the hyperbola:
$9x^2-16y^2=144$ ..... (1)

Equation (1) can be rewritten in the following way:
$$\begin{gathered} \frac{9x^2}{144}-\frac{16y^2}{144}=\frac{144}{144} \\ \Rightarrow \frac{x^2}{16}-\frac{y^2}{9}=1 \end{gathered}$$
$\Rightarrow \frac{x^2}{4^2}-\frac{y^2}{3^2}=1$
This becomes the standard equation of the hyperbola with its major axis $a=4$ and minor axis $b=3$.
Eccentricity, $e=\frac{\sqrt{a^2+b^2}}{a}$
Substituting the value of a and b, we get:
$$\begin{gathered} e=\frac{\sqrt{4^2+3^2}}{4} \\ =\frac{\sqrt{16+9}}{4} \\ =\frac{5}{4} \end{gathered}$$
Therefore, the eccentricity is $\frac{5}{4}$.