General equation of conics

A general second degree equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a conic.


ConditionTypes of conics
Non-degenerateDegenerate
ab - h2 < 0hyperbolapair of intersecting lines
ab - h2 = 0parabolapair of parallel lines or empty set
ab - h2 > 0ellipsepoint or empty set

The equation of the tangent to the conic ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 at a point P(x1,y1) lying on the conic is axx1 + h(xy1 + x1y) + byy1 + g(x + x1) + f(y + y1) + c = 0.



Comments