Question

If the gradient of the tangent at any point \( (x, y) \) of a curve passing through the point \( (1, \frac{\pi}{4}) \) is \[ \left| \frac{dy}{dx} \right| = \frac{1}{x} \cdot \left| \log \left( \frac{y}{x} \right) \right| \] then the equation of the curve is

• A. \( y = \cot(\log x) \)
• B. \( y = \cot(\log x) \)
• C. \( y = \cot(\log x) \)
• D. \( y = \cot(\log x) \)

Hint:

To solve gradient-based problems, differentiate and integrate to find the equation of the curve.

Answer 1

The Correct Option is A

Step 1: Analyzing the gradient.
The given expression for the gradient of the tangent involves both \( x \) and \( y \). Using the relationship \( \frac{dy}{dx} = \frac{1}{x} \), we integrate the equation to find the equation of the curve.

Step 2: Conclusion.
The equation of the curve is \( y = \cot(\log x) \), corresponding to option (1).