Question

The angle made by the tangent at \(\theta=\dfrac{\pi}{3}\) on the curve \(x=a(\theta+\sin\theta)\), \(y=a(1-\cos\theta)\) with \(x\)-axis is

• A. \(\dfrac{\pi}{3}\)
• B. \(\dfrac{\pi}{6}\)
• C. \(\dfrac{2\pi}{3}\)
• D. \(\dfrac{5\pi}{6}\)

Hint:

For parametric curves, always use \[ \frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \] and then compare the slope with \(\tan\alpha\), where \(\alpha\) is the angle made by the tangent with the \(x\)-axis.

Answer 1

The Correct Option is B

Step 1: Differentiate x and y with respect to the parameter θ.
x = a(θ + sin θ) → dx/dθ = a(1 + cos θ). y = a(1 - cos θ) → dy/dθ = a sin θ.

Step 2: Compute the slope of the tangent.
dy/dx = (a sin θ)/[a(1 + cos θ)] = sin θ/(1 + cos θ) = tan(θ/2).

Step 3: Evaluate at θ = π/3.
tan(θ/2) = tan(π/6) = 1/√3. The tangent makes an angle α with the x-axis where tan α = 1/√3 → α = π/6.

Step 4: Final conclusion.
The angle made by the tangent with the x-axis is π/6.