Matrix Representation of Conic Sections - Tangents

Tangents

Through a given point, P, there are generally two lines tangent to a conic. Expressing P as a column vector, p, the two points of tangency are the intersections of the conic with the line whose equation is

When P is on the conic, the line is the tangent there. When P is inside an ellipse, the line is the set of all points whose own associated line passes through P. This line is called the polar of the pole P with respect to the conic.

Just as P uniquely determines its polar line (with respect to a given conic), so each line determines a unique P. This is thus an expression of geometric duality between points and lines in the plane.

As special cases, the center of a conic is the pole of the line at infinity, and each asymptote of a hyperbola is a polar (a tangent) to one of its points at infinity.

Using the theory of poles and polars, the problem of finding the four mutual tangents of two conics reduces to finding the intersection of two conics.