Question

From any point on a hyperbola, tangents are drawn to another hyperbola having the same asymptotes. Show that the chord of contact cuts off a triangle of constant area from the asymptotes.

Answer 1

Let the hyperbolas be

x²/a² - y²/b² = 1  ...(1)

and  x²/a² - y²/b² = k  ...(2)

Let Pa b ( secθ , tanθ) Q Q be a point on the hyperbola provided in Eq. (1). The chord of contact of P with hyperbola provided in Eq. (2) is

x/a secθ - y/b tan θ = k ...(3)

The asymptotes are

y = ±(b/a)x

These asymptotes meet the line provided in Eq. (3) at the points

and hence the area of the triangle cut off is given by

which is constant.