2D curve with two parameters to single parameter

I have been thinking about the following problem. I have a curve in 2D space (x,y), described by the following equation: $a{x}^{2}+bxy+c{y}^{2}+d=0$ where a,b,c,d are known. It is obvious that it is a 1D curve embedded in a 2D space. So I would think there could be such a description of the curve, where only single parameter is present.

It is obvious that you can plug x and then solve a quadratic equation for y, but that is not what I'm looking for. The solution I expect is in the form $x={f}_{x}(t),y={f}_{y}(t),t\in ?$

which can be used in the case of circle equation with sine and cosine of angle.

My goal is to plot the curve in python, so I would like to start at some point of the curve and trace along it. Could you point me to a solution or some materials which are dedicated for such problems?

I have been thinking about the following problem. I have a curve in 2D space (x,y), described by the following equation: $a{x}^{2}+bxy+c{y}^{2}+d=0$ where a,b,c,d are known. It is obvious that it is a 1D curve embedded in a 2D space. So I would think there could be such a description of the curve, where only single parameter is present.

It is obvious that you can plug x and then solve a quadratic equation for y, but that is not what I'm looking for. The solution I expect is in the form $x={f}_{x}(t),y={f}_{y}(t),t\in ?$

which can be used in the case of circle equation with sine and cosine of angle.

My goal is to plot the curve in python, so I would like to start at some point of the curve and trace along it. Could you point me to a solution or some materials which are dedicated for such problems?