A parabola has one line of symmetry. The line contains the vertex and focus of the parabola and goes in the same direction that the parabola opens. (It is perpendicular to the directrix).

A circle has infinitely many lines of symmetry. Any line that contains a diameter of the circle is a line of symmetry.

An ellipse has two lines of symmetry. One contains the major axis and the other contains the minor axis. They are perpendicular to each other and intersect at the center of the ellipse.

A hyperbola has two lines of symmetry. One contains the transverse axis and the other contains the conjugate axis. They are perpendicular to each other and intersect at the center of the hyperbola.