Question

The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all ______.

Options

• parallel to x-axis

• parallel to y-axis

• parallel to the line y = x

• parallel to the line y = –x

Answer 1

The normals to the curve x = a(θ + sinθ), y = a(1 – cosθ) at the points θ = (2n + 1)π, n∈I are all parallel to y-axis.

Explanation:

x = a(θ + sinθ)

y = a(1 – cosθ)

Differentiating above functions, w.r.t.θ, we get

`{:((dx)/(dθ) = a(1 + cosθ)),((dy)/(dθ) = a(sinθ)):}}`

and `(dy)/(dx) = ((dy)/(dθ))/((dx)/(dθ))`

= `(asinθ)/(a(1 + cosθ))`

= `sinθ/(1 + cosθ)`

`(dy)/(dx) = sinθ/(1 + cosθ)`

θ = (2n + 1)π, n∈I 

`(dy)/(dx) = sinθ/(1 + cosθ)` is undefined

For all θ = (2n + 1)π in n∈I 

Hence, normal must be parallel to y-axis.