Question

Answer the following:

Find the equation of the tangent to the hyperbola x = 3 secθ, y = 5 tanθ at θ = `pi/3`

Answer 1

Given, equation of the hyperbola is

x = 3 sec θ, y = 5 tan θ.

Since sec²θ  –  tan²θ = 1,

`x^2/9 - y^2/25` = 1

Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get

a² = 9 and b² = 25

∴ a = 3 and b = 5

Equation of tangent at P(θ) is

`(xsectheta)/"a" - (ytantheta)/"b"` = 1

∴ Equation of tangent at P`(pi/3)` is

`(xsec(pi/3))/3 - (ytan(pi/3))/5` = 1

∴ `(2x)/3 - (sqrt(3)y)/5` = 1

∴ `10x - 3sqrt(3)y` = 15