Question

The family of curves in which the sub tangent at any point of a curve is double the abscissae, is given by
(a) x = Cy²
(b) y = Cx²
(c) x² = Cy²
(d) y = Cx

Answer 1

(a) x = Cy²

$\mathrm{Subtangent}=\frac{y}{\frac{dy}{dx}}$

It is given that subtangent at any point of a curve is double of the abscissa.

$\begin{array}{l}\therefore \frac{y}{\frac{dy}{dx}}=2x\\ y=2x\frac{dy}{dx}\\ \int \frac{dx}{x}=2\int \frac{dy}{y}\\ \ln x=2\ln y+a\\ \ln x=\ln y^2+\ln c\\ \ln x=\ln c{y}^2\\ x=c{y}^2\end{array}$