Question

The portion of the tangent to a hyperbola intercepted between its asymptotes is bisected at the point of contact.

Answer 1

Let P(a secθ , b tanθ) be a point on

x²/a² - y²/b² = 1

The tangent at P is

xsecθ/a - ytanθ/b = 1  ....(1)

We know that the equations of the asymptotes are

y = ±(b/a) x

Substituting y = (b/a) x in Eq. (1), we get

 

Now, it is easy to see that the midpoint of QR is P(a secθ , b tanθ).