Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
x = a (θ – sin θ), y = a(1 + cos θ) at

θ = – π/2

Question

Find the The Slopes of the tangent and the normal to the following curves at the indicated points : x = a (θ – sin θ), y = a(1 + cos θ) at θ = – π/2

Philoid · Accepted Answer

Given:

x = a() & y = a(1 + cos) at

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

(xⁿ) = n.x^{n – 1}

⇒ x = a()

⇒ = a(() – (sin))

⇒ = a(1 – ) ...(1)

(sinx) = cosx

⇒ y = a(1 + cos)

⇒ = a(() + (cos))

(cosx) = – sinx

(Constant) = 0

⇒ = a( + ( – sin))

⇒ = a( – sin)

⇒ = – asin ...(2)

⇒

The Slope of the tangent is

Since,

⇒

sin() = 1

cos() = 0

⇒

⇒ = 1

The Slope of the tangent at x = is 1

⇒ The Slope of the normal =

⇒ The Slope of the normal = – 1

Find the The Slopes of the tangent and the normal to the following curves at the indicated points :x = a (θ – sin θ), y = a(1 + cos θ) atθ = – π/2