If x = a cos θ , y = b sin θ, show 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 cos θ ……equation 1
y = b sin 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.
…..equation 3
Similarly, ……equation 4
[∵
Differentiating again w.r.t x :
…..equation 5
[ using chain rule and ]
From equation 3:
∴
Putting the value in equation 5 :
From equation 1:
y = b sin θ
∴ …..proved.