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

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