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

The correct option is rectangular hyperbola.

Let the coordinates of any point on the curve be represented by $(x,y)$, where $x$ represents the abscissa and $y$ represents the ordinate.
The slope of the tangent to the curve at any point $(x,y)$ is given by the derivative:
$\frac{dy}{dx}$

According to the given condition, the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of that point. Mathematically, this can be written as:
$\frac{dy}{dx}=\frac{x}{y}$

To find the equation of the curve, we can solve this differential equation by separating the variables:
$ydy=xdx$

Integrating both sides of the equation gives:
$\int ydy=\int xdx$
$\frac{y^2}{2}=\frac{x^2}{2}+C$
where $C$ is the constant of integration.

Multiplying the entire equation by 2, we get:
$y^2=x^2+2C$
Letting $2C=K$ (another constant), we can rearrange the terms as:
$x^2-y^2=-K$

This equation is of the form $x^2-y^2=a{b}^{}$ (or $x^2-y^2=\text{constant}$), which represents a rectangular hyperbola.