Question

If the length of the sub-tangent at any point P on a curve is proportional to the abscissa of the point P, then the equation of that curve is (C is an arbitrary constant):

• A. \( y^k + x^k = C \)
• B. \( x^{1/k} C = y^k \)
• C. \( (x + y)^k = C \)
• D. \( y = x^{1/k} C \)

Hint:

Use differential equations and separation of variables to solve problems involving proportional relationships.

Answer 1

The Correct Option is D

Step 1: Use the property of sub
-tangent The length of the sub
-tangent is given by: \[ \frac{y}{\frac{dy}{dx}}. \] Since it is proportional to the abscissa \( x \), \[ \frac{y}{\frac{dy}{dx}} = kx. \] Step 2: Solve the differential equation Rearranging, \[ \frac{dy}{dx} = \frac{y}{kx}. \] Separating variables, \[ \frac{dy}{y} = \frac{dx}{kx}. \] Integrating both sides, \[ \ln y = \frac{1}{k} \ln x + C. \] Taking exponentials, \[ y = x^{1/k} C. \]