If x = a (θ + sin θ), y = a (1+ cos θ), prove that .
Idea of parametric form of differentiation:
If y = f (θ) and x = g(θ) i.e. y is a function of θ and x is also some other function of θ.
Then dy/dθ = f’(θ) and dx/dθ = g’(θ)
We can write :
Given,
x = a (θ + sin θ) ……equation 1
y = a (1+ cos θ) ……equation 2
to prove :
We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.
Let’s find
As,
So, lets first find dy/dx using parametric form and differentiate it again.
[∵ from equation 2] …..equation 3
Similarly,
……equation 4
[∵
[∵ from equation 2] …..equation 5
Differentiating again w.r.t x :
Using product rule and chain rule of differentiation together:
[using equation 3 and 5]
[ from equation 1]
∴ ….proved