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.xn – 1
⇒ x = acos3
⇒ = a((cos3))
(cosx) = – sinx
⇒ = a(3cos3 – 1 – sin)
⇒ = a(3cos2 – sin)
⇒ = – 3acos2sin ...(1)
⇒ y = asin3
⇒ = a((sin3))
(sinx) = cosx
⇒ = a(3sin3 – 1cos)
⇒ = 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