Question

If x = a cos³θ, y = a sin³θ, then `1 + (("d"y)/("d"x))^2` is ______.

Options

• sec³θ

• tan θ

• 1

• tan²θ

Answer 1

If x = a cos³θ, y = a sin³θ, then `1 + (("d"y)/("d"x))^2` is sec³θ.

Explanation:

x = a cos³θ and y = a sin³θ

∴ `("d"x)/("d"theta) = - 3"a"cos^2theta.sin theta`

and `("d"y)/("d"theta)` = 3a sin²θ.cos θ

∴ `("d"y)/("d"theta) = - tantheta`

∴ `1 + (("d"y)/("d"x))^2 = 1 + tan^2theta` = sec³θ