A line has a linear equation. A conic has a quadratic equation. It follows that a line and a conic meet at most twice. If the equations result in two coincident roots, then we say that the line is a tangent to the conic at the point where they meet.
For the general conic, obtaining tangents algebraically is tedious. We shall look The Parabola P : y^{2} = x.
(1) The line y = k cuts P exactly once.
Note that each family {L : y = nx+c, c real} of parallel lines contains
Although we are used to the hyperbola as x^{2}y^{2}=1, we can rotate through the The Hyperbola H : xy = 1.
(1) The lines x = 0 and y = 0 do not meet H.
Note that the family {L : y = mx+c, c real} contains
Each contains one line which does not meet H  these are the asymptotes  while the other lines cut H exactly once, but are not tangents.
Although we do not regard the circle as a euclidean conic, the results for the The Circle
For any circle C, each family of parallel lines contains exactly two tangents This generalises to ellipses, as we shall see when we look at affine geometry.

