Find the The Slopes of the tangent and the normal to the following curves at the indicated points :

x = a cos³ θ, y = a sin³ θ at θ = π/4

Given:

x = acos³ & y = asin³ at

Here, To find , we have to find & and and divide and we get our desired .

(xⁿ) = n.x^{n – 1}

⇒ x = acos³

⇒ = a((cos³))

(cosx) = – sinx

⇒ = a(3cos^{3 – 1} – sin)

⇒ = a(3cos² – sin)

⇒ = – 3acos²sin ...(1)

⇒ y = asin³

⇒ = a((sin³))

(sinx) = cosx

⇒ = a(3sin^{3 – 1}cos)

⇒ = a(3sin²cos)

⇒ = 3asin²cos ...(2)

⇒

⇒

⇒ = – tan

The Slope of the tangent is – tan

Since,

⇒ = – tan()

⇒ = – 1

tan() = 1

The Slope of the tangent at x = is – 1

⇒ The Slope of the normal =

⇒ The Slope of the normal =

⇒ The Slope of the normal =

⇒ The Slope of the normal = 1