Let C be a curve defined parametrically as x = a cos³ θ, y = a sin³ θ, 0 ≤θ ≤ π/2. Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a). (CBSE 2014)

sin x and cos x functions are continuous everywhere on (−∞, ∞) and differentiable for all arguments.

So both the necessary conditions of Lagrange’s mean value theorem is satisfied.

x = a cos³ θ

y = a sin³ θ

We know that,

sin² θ + cos² θ = 1

x = acos³ θ

y = asin³ θ

For f’(c), put the value of x=c in f’(x):

f’(c) = – tan θ

For f(a), put the value of x=a in f(x):

= 0

For f(0), put the value of x=0 in f(x):

= a

⇒ – tan θ = – 1

⇒ tan θ = 1

Now put the value of θ in the function of x and y:

x = a cos³ θ

Similarly,

y = a sin³ θ