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. $\sqrt{\frac{5}{2}}$
• B. $\sqrt{\frac{3}{2}}$
• C. $\sqrt{2}$
• D. $\sqrt{2}$

Answer 1

The correct option is B $\sqrt{\frac{3}{2}}$

$Normalat(6,3)is\frac{a^{2}x}{6}-\frac{b^{2}y}{3}=a^2+b^2,\Rightarrow \frac{9a^2}{6}=a^2+b^2\Rightarrow \frac{3}{2}=1+\frac{b^2}{a^2}\therefore \frac{b^2}{a^2}=\frac{1}{2}\Rightarrow e^2-1=\frac{1}{2}\Rightarrow e=\sqrt{\frac{3}{2}}$