Question

Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ If the normal at the point $P$ intersects the $x-$axis at $(9,0)$, then the eccentricity of the hyperbola is

• A. $\sqrt{3}$
• B. Option c equation is not clear
• C. Option a equation is not clear
• D. Option b equation is not clear

Answer 1

The correct option is D Option b equation is not clear


The equation of the normal at $P(6,3)$ to the given hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
$\frac{a^{2}x}{6}+\frac{b^{2}y}{3}=a^2+b^2$
which meets $x-$axis at $(9,0)$, hence
$\frac{9a^2}{6}=a^2+b^2$
$\Rightarrow \frac{3}{2}=1+\frac{b^2}{a^2}=e^2$
$\Rightarrow e=\sqrt{\frac{3}{2}}$