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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 ______________.

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Solution

## 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 $\frac{dy}{dx}=\frac{x}{y}\phantom{\rule{0ex}{0ex}}⇒ydy=xdx\phantom{\rule{0ex}{0ex}}\mathrm{Integrating}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}\phantom{\rule{0ex}{0ex}}⇒\int ydy=\int xdx\phantom{\rule{0ex}{0ex}}⇒\frac{{y}^{2}}{2}=\frac{{x}^{2}}{2}+C\phantom{\rule{0ex}{0ex}}⇒{y}^{2}={x}^{2}+2C\phantom{\rule{0ex}{0ex}}⇒{y}^{2}={x}^{2}+A,\mathrm{where}A=2C\mathrm{is}\mathrm{arbitrary}\mathrm{constant}\phantom{\rule{0ex}{0ex}}⇒{y}^{2}-{x}^{2}=A\phantom{\rule{0ex}{0ex}}$ 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 $\overline{){y}^{2}-{x}^{2}=A}.$

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