Question

The equation to the curve, which is such that portion of the axis of x cut off between the origin and the tangent at any point is proportional to the ordinate of that point, is

(k is constant of proportionality)

• A. x = y (c − k log y)
• B. log x = ky² + c
• C. x² = y (c − k log y)
• D. None of these

Answer 1

Answer: (A)

x = y (c − k log y)

Answer 2

Let the curve be y = f (x). The equation of the tangent at any point (x, y) is given by Y − y = f′ (x) (X − x). So the portion of the axis of X which is cut off between the origin and the tangent at any point is obtained by putting Y = 0. Therefore

x − $\frac{y}{f^{'}(x)}$ = ky

⇒ x − y $\frac{dx}{dy}$ = ky

⇒ $\frac{dx}{dy}$ − $\frac{x}{y}$ = − k

which is a linear equation in x, and its integrating factor is

$e^{- \int_{}^{}{1/ydy}}$ = y⁻¹.

Therefore, multiplying by y⁻¹ we have

$\frac{d}{dy}$ (xy⁻¹) = − ky⁻¹

⇒ xy⁻¹ = − k log y + c

or x = y (c − k log y)