Find \(\frac{dy}{dx}\) if x = a sec θ and y = b tan θ ?

Question

Question

Answer 1

CONCEPT:

If x = f(t), y = g(t), where t is a parameter, then \(\frac{{dy}}{{dx}} = \frac{{g'\left( t \right)}}{{f'\left( t \right)}}\)

CALCULATION:

Given: x = a sec θ and y = b tan θ

Here, we have to find \(\frac{dy}{dx}\)

So, first we have to find dx/dθ and dy/dθ

As we know that, \(\frac{{d\left( {\sec x} \right)}}{{dx}} = \sec x ⋅ tan \ x\)

⇒ dx/dθ = a sec θ ⋅ tan θ------(1)

As we know that, \(\frac{{d\left( {\tan x} \right)}}{{dx}} = \; \sec^2 x\)

⇒ dy/dθ = b sec² θ-----------(2)

From (1) and (2), we can say that, \(\frac{dy}{dx} = \frac{b}{a} cosec \ \theta\)

Hence, correct option is 3.