Question

Show that the product of lengths of perpendicular segments drawn from the foci to any tangent to the hyperbola `x^2/25 + y^2/16 = 1` is equal to 16.

Answer 1

Equation of the hyperbola is `x^2/25 + y^2/16 = 1`

Here, a² = 25, b² = 16

∴ a = 5, b = 4

∴ e = `sqrt(a^2 + b^2)/a = sqrt41/5`

∴ ae = `5(sqrt41/5) = sqrt41`

∴ Foci are S(ae, 0) ≡ S(`sqrt41` , 0) and 

and S'(−ae, 0) ≡ S'(− `sqrt41` , 0)

Equation of tangent to the hyperbola with slope m is

y = mx + `sqrt(a^2m^2 − b^2)`

∴ y = mx + `sqrt(25m^2 - 16) = 0` ....(i)

p₁ = length of perpendicular segment from the focus S(`sqrt41` , 0) to the tangent (i)