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  4. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

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Q. The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is

Differential Equations

Solution:

Slope of tangent, $\frac{dy}{dx}=\frac{x}{y}$
$\Rightarrow ydx=xdx$
On integrating, we get
$\frac{y^{2}}{2}=\frac{x^{2}}{2}+k$
$\Rightarrow y^{2}-x^{2}=2k=c$ (say)
which is equation of a rectangular hyperbola.