Question

If the angle between the curves \(y^2=4x\) and \(y=e^{-x/2}\) is \(\theta\), then \(\cosec^2\left(\dfrac{\theta}{2}\right)\) is

• A. \(2\)
• B. \(3\)
• C. \(\sqrt{3}\)
• D. \(\sqrt{2}\)

Hint:

If \[ m_1m_2=-1, \] then the tangents are perpendicular and the angle between the curves is \[ 90^\circ. \] This often simplifies trigonometric calculations involving the angle between curves.

Answer 1

The Correct Option is A

Step 1: Locate the intersection point of the two curves.
y² = 4x and y = e^(-x/2). At x = 0, y = e⁰ = 1, which satisfies the second curve. The point (0, 1) is the intersection.

Step 2: Find the slope of each curve at this point.
For the parabola: 2y(dy/dx) = 4 → dy/dx = 2/y. At (0, 1): m₁ = 2. For the exponential: dy/dx = -(1/2)e^(-x/2). At x = 0: m₂ = -1/2.

Step 3: Compute the angle between the tangents.
tan θ = |(m₁ - m₂)/(1 + m₁m₂)| = |(2 + 1/2)/(1 - 1)| → denominator zero → tan θ = ∞ → θ = π/2.

Step 4: Evaluate cosec²(θ/2).
θ/2 = π/4. cosec²(π/4) = (1/sin(π/4))² = (√2)² = 2.

Step 5: Final conclusion.
cosec²(θ/2) = 2.