Question

The slope of the tangent at a point P (x, y) on a curve is $\frac{-x}{y}$. If the curve passes through the point (3, −4), find the equation of the curve.

Answer 1

$$\begin{gathered} \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \frac{dy}{dx}=\frac{-x}{y} \\ \Rightarrow ydy=-xdx \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ \int ydy=-\int xdx \\ \Rightarrow \frac{y^2}{2}=-\frac{x^2}{2}+C \\ \mathrm{Since}\mathrm{the}\mathrm{curve}\mathrm{passes}\mathrm{through}\left(3,-4\right),\mathrm{it}\mathrm{satisfies}\mathrm{the}\mathrm{above}\mathrm{equation}. \\ \therefore \frac{{\left(-4\right)}^2}{2}=-\frac{3^2}{2}+C \\ \Rightarrow 8=-\frac{9}{2}+C \\ \Rightarrow C=\frac{25}{2} \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}C,\mathrm{we}\mathrm{get} \\ \frac{y^2}{2}=-\frac{x^2}{2}+\frac{25}{2} \\ \Rightarrow x^2+y^2=25 \end{gathered}$$