Let x(t)=2 √2 cos t √(sin 2 t) and
y(t)=2 √2 sin t √(sin 2 t), t ∈(0, π/2).
Then (1+((d y)/(d x))²)/((d² y)/(d x²)) at t=π/4 is equal to :

Solution

Maths sir · 4 steps

Take a common √(sin 2t) in each derivative and dy/dx collapses to just tan 3t.

1. First derivatives. Use the product rule, then put both terms over √(sin 2t). The top of dx/dt becomes cos t cos 2t - sin t sin 2t = cos 3t. The top of dy/dt becomes sin t cos 2t + cos t sin 2t = sin 3t.

   dx/dt = (2√2 cos 3t)/√(sin 2t), dy/dt = (2√2 sin 3t)/√(sin 2t) ⇒ dy/dx = tan 3t

2. Second derivative. Differentiate tan 3t with respect to t, then divide by dx/dt. This gives d²y/dx² = (3 sec² 3t)/(dx/dt).
3. Ratio simplifies. The numerator is 1 + tan² 3t = sec² 3t, which cancels with the sec² 3t in the second derivative. The whole expression becomes 1/3·dx/dt = (2√2 cos 3t)/3√(sin 2t).
4. So. At t = π/4, sin 2t = 1 and cos 3π/4 = -1/√2. The value is 2√2/3×(-1/√2) = -2/3, which is option (d).

The idea to keep. In parametric questions, simplify dx/dt and dy/dt with trig identities before dividing. Remember that d²y/dx² is d/dt(dy/dx) divided by dx/dt.

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