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

• 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: (B)

log x = Ky² + C

Answer 2

[K is constant of proportionality] Let the curve be y = f(x). The equation of 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 Ž x – y $\frac{dx}{dy}$= Ky Ž $\frac{dx}{dy}$ – $\frac{x}{y}$= – K

which is a linear equation in x, so its integrating factor is $e^{- \int_{}^{}{(1/y)dy}}$= y^–1. Therefore, multiplying by y^–1, we have

$\frac{d}{dy}$ (xy^–1) = – Ky^–1 Ž xy^–1 = –K log y + C

Ž x = y (C –K log y)

where C is arbitrary constant