Question

The length of the tangent drawn at the point \( P \left( \frac{\pi}{4} \right) \) on the curve \( x^{2/3} + y^{2/3} = 2^{2/3} \) is:

• A. \( \frac{2}{3} \)
• B. \( 1 \)
• C. \( \frac{4}{3} \)
• D. \( 2 \)

Hint:

For the length of the tangent, use implicit differentiation and apply the standard tangent length formula.

Answer 1

The Correct Option is B

Step 1: Differentiate the given curve equation The given equation of the curve is: \[ x^{2/3} + y^{2/3} = 2^{2/3}. \] Differentiating both sides with respect to \( x \): \[ \frac{2}{3} x^{-1/3} + \frac{2}{3} y^{-1/3} \frac{dy}{dx} = 0. \] Rearranging for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = -\frac{x^{-1/3}}{y^{-1/3}}. \] Step 2: Compute the slope at the given point Let the given point be \( P(x_0, y_0) \). To determine \( x_0 \) and \( y_0 \), we use the constraint \( x^{2/3} + y^{2/3} = 2^{2/3} \) with \( x_0 = \frac{\pi}{4} \). Solving for \( y_0 \), we find its corresponding value. Step 3: Use the formula for the length of the tangent The formula for the length of the tangent to a curve at a given point is: \[ L = \frac{|x_0 dy/dx + y_0 - f(x_0, y_0)|}{\sqrt{(dy/dx)^2 + 1}}. \] Substituting the computed values, we get: \[ L = 1. \]