Question

The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:

• A. \(1\)
• B. \(\frac{1}{3}\)
• C. \(\frac{1}{2}\)
• D. \(0\)

Hint:

In problems involving differential calculus, particularly those involving curve properties like tangents and normals, remember to simplify derivative expressions where possible and analyze the conditions under which terms become constant.

Answer 1

The Correct Option is C

To find the value of \( k \) for which the subnormal is constant, we use the formula for the subnormal of a curve \( y = f(x) \): \[ \text{Subnormal} = y \cdot \frac{dy}{dx} \quad \text{at any point}. \] For the given function: \[ y = \left(\frac{x}{2024}\right)^k. \] The first derivative of \( y \) with respect to \( x \) is: \[ \frac{dy}{dx} = k \left(\frac{x}{2024}\right)^{k-1} \cdot \frac{1}{2024}. \] Now, the subnormal at any point is given by: \[ \text{Subnormal} = y \cdot \frac{dy}{dx} = \left(\frac{x}{2024}\right)^k \cdot k \left(\frac{x}{2024}\right)^{k-1} \cdot \frac{1}{2024}. \] Simplifying this: \[ \text{Subnormal} = k \cdot \left(\frac{x}{2024}\right)^{2k-1} \cdot \frac{1}{2024}. \] For the subnormal to be constant, the expression should not depend on \( x \). This happens when the power of \( x \) in the expression is 0, i.e., when: \[ 2k - 1 = 0 \quad \Rightarrow \quad k = \frac{1}{2}. \]