Question

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{5}{4}$ and $2x+3y-6=0$ is a focal chord of the hyperbola, then the length of the transverse axis is

• A. $\frac{12}{5}$
• B. $16$
• C. $\frac{24}{7}$
• D. $\frac{24}{5}$
• E. $\frac{12}{7}$

Answer 1

The correct option is D

$\frac{24}{5}$

Explanation for correct option

Given that the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{5}{4}$

That is eccentricity $e=\frac{5}{4}$

The focus of hyperbola is $\left(\pm ae,0\right)$

$2x+3y-6=0$ is the focal chord of the hyperbola

For $\left(ae,0\right)$ (Considering one focus only)

$$\begin{gathered} \Rightarrow 2\left(ae\right)+3·0-6=0 \\ \Rightarrow ae=3 \\ \Rightarrow a=\frac{3}{e} \\ \Rightarrow a=\frac{12}{5} \end{gathered}$$

$\therefore$The length of the transverse axis $=2a=\frac{24}{5}$

Hence, Option (D) is correct i.e. $\frac{24}{5}$