Let P(6, 3) be a point on hyperbola x2a2−y2b2=1. If the normal at the point P intersect the x - axis at (9, 0), then the eccentricity of the hyperbola is
(IIT JEE 2011)
√32
We have the equation of hyperbola
x2a2−y2b2=1
Equation of normal at point (x1,y1) is
a2xx1−b2yy2=a2+b2
Equation of normal at point (6, 3)
a2x6−b2y3=a2+b2
Equation of normal at point (9, 0)
9a26=a2+b2
⇒3a22−a2=b2
⇒3a22a2−a2a2=b2a2
⇒32−1=b2a2
⇒b2a2=12
⇒a2=2b2
Let the eccentricity of the hyperbola be e.
e2=1+b2a2
e2=1+12
⇒e2=32
⇒e=√32