S = ax²+by²+2gx+2fy+2hxy+c = 0

Partially differentiate S w.r.t x and y respectively.

2ax + 2g + 2hy = 0

ax+hy+g = 0     ...............(i)

2by + 2f + 2hx = 0

by + hx + f = 0    ..............(ii)

Solve (i) and (ii) for x and y.

(x,y) is the point of intersection.

Any similar method (generic) for intersection of two curves/ a curve and a line?

This method isnt actually to find the intersection of two curves. This is used to find the centre of the curve. Incase of pair of straight lines, it is infact, the point of their intersection. In ellipse and hyperbola, u get the center.