If x = 2 sin θ - sin 2θ and y = 2 cos θ - cos 2θ,
θ ∈ [ 0,2π ], then d²y/dx² at θ = π is :

Solution

Maths sir · 4 steps

Simplify dy/dx to a single cot first, then remember to divide by dx/dθ once more.

1. First derivative. dx/dθ = 2 cos θ - 2 cos 2θ and dy/dθ = -2 sin θ + 2 sin 2θ. So dy/dx = (sin 2θ - sin θ)/(cos θ - cos 2θ).
2. Sum to product. The top is 2 cos 3θ/2 sin θ/2 and the bottom is 2 sin 3θ/2 sin θ/2. The sin θ/2 cancels, leaving dy/dx = cot 3θ/2.

   d²y/dx² = (d/dθ(dy/dx))/(dx/dθ) = (-3/2 cosec²3θ/2)/(2 cos θ - 2 cos 2θ)

3. Put in values. At θ = π, cosec²3π/2 = 1, so the top is -3/2. The bottom is 2(-1) - 2(1) = -4.
4. So. d²y/dx² = (-3/2)/-4 = 3/8, which is option (a).

The idea to keep. For parametric curves, the second derivative is the θ-derivative of dy/dx divided by dx/dθ. It is never just the ratio of the two second derivatives.

Answer from the NTA final answer key. Solution by PrepWiser.