What is the point of contact between the hyperbola
x2a2−y2b2=1 and
the tangent y=mx±√a2m2−b2.
[±a2m√a2m2−b2,±b2√a2m2−b2]
Solving the equation of hyperbola and tangent will give the point of contact.
x2a2−(mx±√a2m2−b2)2b2=1
b2x2−a2(m2x2+a2m2−b2±2mx√a2m2−b2=a2b2
∴ x=±2ma2√a2m2−b2±√4m2a4(a2m2−b2)−4(b2−a2m2)(−a4m2)2(b2−a2m2)
=±2ma2[√a2m2−b2±√a2m2−b2+b2−a2m2]2(b2−a2m2)
=±a2m√a2m2−b2(b2−a2m2)
=∓a2m√a2m2−b2
⇒y=∓b2√a2m2−b2
∴ The required point of contact is
(∓a2m√a2m2−b2,∓b2√a2m2−b2)