Question

P is a point on the hyperbola  x²/a² - y²/b² = 1 The tangent at P meets the asymptotes at Q and R. If C is the centre of the curve, then show that  CP. CQ = a² + b²

Answer 1

Let P = (a secθ , b tanθ)  so that the tangent at P is

and x/a secθ - y/btanθ = 1   ....(1)

and  the two asymptotes are

x/a - y/b = 0  .....(2)

and   x/a + y/b = 0   ....(3)

Solving Eqs. (1) and (2), we have

Q = [a(secθ + tanθ), b(secθ + tanθ)]

and solving Eqs. (1) and (3), we have

Hence, CP. CQ = a² + b²