Let `P(6, 3)` be a point on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1.` If the normal at point P intersects the x-axis at (9, 0), then find the eccentricity of the hyperbola.

Options

A. `sqrt((5)/(2))`
B. `sqrt((3)/(2))`
C. `sqrt(2)`
D. `sqrt(3)`

Explanation

The equation of the normal to `(x^(2))/(a^(2))-(y^(2))/(b^(2))-1` at `P(6,3)` is
`(a^(2)x)/(6)-(b^(2)y)/(3)=a^(2)+b^(2)`
The cuts `x-`axis at `(9,0)`. Therefore,
`(3a^(2))/(2)=a^(2)+b^(2)implies(3a^(2))/(2)=a^(2)e^(2)impliese=sqrt((3)/(2))`