Question

The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa and ordinate of the point, is ______________.

Answer 1

Given: Slope of the tangent at any point is equal to the ratio of the abscissa and ordinate of the point.

According to the question
$$\begin{gathered} \frac{dy}{dx}=\frac{x}{y} \\ \Rightarrow ydy=xdx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \int ydy=\int xdx \\ \Rightarrow \frac{y^2}{2}=\frac{x^2}{2}+C \\ \Rightarrow y^2=x^2+2C \\ \Rightarrow y^2=x^2+A,\mathrm{where}A=2C\mathrm{is}\mathrm{arbitrary}\mathrm{constant} \\ \Rightarrow y^2-x^2=A \end{gathered}$$

which is the equation of a rectangular hyperbola.

Hence,the curve for which the slope of the tangent at any point is equal to the ratio of the abscissa and ordinate of the point, is $\underline{y^2-x^2=A}.$