Let's consider a second degree equation in two variables say, $$ S\equiv ax^2+by^2+2hxy+2gx+2fy+c=0 $$ where $a, b, c, f, g$ and $h$ are constants. To be clear, I'm dealing with conic sections here.
Depending on given data, the above equation can represent any of the following, namely
- pair of intersecting straight lines
- circle
- parabola
- ellipse
- hyperbola
Let's now consider the following two equations
$$L_1\equiv \frac{\partial S}{\partial x}=2ax+2hy+2g=0 \\ L_2\equiv \frac{\partial S}{\partial y}=2by+2hx+2f=0$$
It is widely known that if
The equation represents pair of straight line: The intersection of $L_1$ and $L_2$ give the point of intersection of the lines.
The equation represents a circle: The intersection of $L_1$ and $L_2$ give the centre of the circle.
The equation represents a parabola: The lines $L_1$ and $L_2$ give us an imaginary intersection. (Analogous with the argument that the parabola is a conic section whose center could be assumed to be taken at infinity).
The equation represents a ellipse: The intersection of $L_1$ and $L_2$ give the centre of the ellipse.
The equation represents a hyperbola: The intersection of $L_1$ and $L_2$ give the centre of the hyperbola.
But how were the above statements proven? Is there a way to graphically visualise what's going on? Why do these statements even make sense?