Question

Find the equation of the tangent and the normal to the given curve at the indicated point: x²/a² - y²/b² = 1 at (a sec θ, b tan θ)

Answer 1

Consider y = x²/a² - y²/b² = 1 as the equation of curve

By differentiating both sides w.r.t. x

By substituting the values

y – b tan θ = – (a tan θ/b sec θ) (x – a sec θ)

On further calculation

by sec θ – b² tan θ sec θ = – ax tan θ + a² sec θ tan θ

By taking sec θ tan θ as common

ax tan θ + by sec θ = sec θ tan θ(a² + b²)

So we get

ax cos θ + by cot θ = a² + b²