Question

At each point $(x,y)$of a curve, the intercept of the tangent on the$y-axis$ is equal to $2x{y}^2$. Find the equation of the curve.

• A. $\frac{x}{y}=x^2+c$
• B. $\frac{y}{x}=y^2+c$
• C. $\frac{x}{y}=y^2+c$
• D. $\frac{y}{x}=x^2+c$

Answer 1

The correct option is A

$\frac{x}{y}=x^2+c$

Given that $y-intercept$ , $c=2x{y}^2$

As the Equation of the tangent is $y=mx+c$

and Slope $m=\frac{dy}{dx}$

So we have,

$$\begin{gathered} y=\left(\frac{dy}{dx}\right)x+2x{y}^2 \\ \Rightarrow ydx-xdy=2x{y}^{2}dx \\ \Rightarrow \frac{(ydx-xdy)}{y^2}=2xdx \\ \Rightarrow \int \frac{(ydx-xdy)}{y^2}=\int 2xdx[integratingbothside] \\ \Rightarrow \int d\left(\frac{x}{y}\right)=\int 2xdx \\ \Rightarrow \frac{x}{y}=x^2+c \end{gathered}$$

Hence, option (a) is the correct option.