Practicing Success
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) |
x = y (c - k log y) log x = ky2 + c x2 = y (c - k log y) none of these |
x = y (c - k log y) |
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). 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{dy}{dx} = ky $ $⇒ \frac{dy}{dx} - \frac{x}{y} = - k$ which is a linear equation in x, and its integrating factor is $e^{-∫1/y\,dx}=y^{-1}$ Therefore, multiplying by y-1 we have $\frac{d}{dx}(xy^{-1}) = - ky^{-1}$ xy-1 = - k log y + c or x = y (c - k log y). Hence (A) is the correct answer. |