The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is :

${\textbf{Step -1: Differentiate slope}}{\textbf{.}}$

${\text{Given: slope of tangent is equal to ratio of abscissa to the ordinate of the point}}{\text{.}}$

${\text{Let slope of the tangent be }}\dfrac{{dy}}{{dx}}.$

$\text{According to the question}$

$\dfrac{{dy}}{{dx}} = \dfrac{x}{y}$

${\textbf{Step -2: Solve further to find the equation}}{\textbf{.}}$

${\text{Separating the variables}}{\text{.}}$

$ydy = xdx$

${\text{Upon integration we get,}}$

$\int {ydy = \int {xdx} }$

$\Rightarrow\dfrac{{{y^2}}}{2} = \dfrac{{{x^2}}}{2} + C$

$\Rightarrow{y^2} = {x^2} + 2C$

$\Rightarrow {x^2} - {y^2} + 2C = 0$

$\Rightarrow {x^2} - {y^2} = k$

$\textbf{[ k = - 2 C]}$

${\textbf{Hence, the given equation is of a rectangular hyperbola}}{\textbf{.}}$