If the angle made by the tangent at the point (x₀, y₀) on the curve x = 12(t + sin t cos t), y = 12(1 + sin t)², 0 < t < π/2, with the positive x-axis is π/3, then y₀ is equal to:

Solution

Maths sir · 3 steps

Find dy/dx in terms of t, set it equal to tan π/3 = √3, and the right t falls out.

1. Differentiate in t. dx/dt = 12(1 + cos² t - sin² t) = 12(1 + cos 2t) = 24 cos² t. Also dy/dt = 24(1 + sin t)cos t.

   dy/dx = (24(1+sin t)cos t)/(24 cos² t) = (1+sin t)/(cos t) = tan(π/4 + t/2)

2. Match the slope. The tangent makes angle π/3 with the x-axis, so tan(π/4 + t/2) = tan π/3. For 0 < t < π/2 this gives t/2 = π/12, so t = π/6. Check: (1 + 1/2)/(√3/2) = 3/√3 = √3.
3. So. y₀ = 12(1 + 1/2)² = 12 × 9/4 = 27. The answer is (c).

The idea to keep. In parametric questions, the given angle fixes the slope dy/dx, not the parameter. Solve for t from the slope first, then substitute that t into x or y.

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