Ellipses

In this section we shall study two important types of curves, the ellipses and hyperbolas. The intersection of a circular cone and a plane will always be either a parabola, an ellipse, a hyperbola, or one of three degenerate cases — one line, two lines, or a point. For this reason, parabolas, ellipses, and hyperbolas are called conic sections. We begin with the definition of an ellipse in the plane.

If the two foci F₁ and F₂ are the same, the ellipse is just the circle with center at the focus and diameter L. Circles are discussed in Section 1.1.

We shall concentrate on the case where the foci F₁ and F₂ are different. The ellipse will be an oval curve shown in Figure 5.5.1. The orbit of a planet is an ellipse with the sun at one focus. The eye sees a tilted circle as an ellipse.

Figure 5.5.1 Ellipse

The line through the foci F₁ and F₂ is called the major axis of the ellipse. The point on the major axis halfway between the foci is called the center. The line through the center perpendicular to the major axis is called the minor axis.

An ellipse is symmetric about both its major and its minor axes. That is, for any point P on the ellipse, the mirror image of P on the other side of either axis is also on the ellipse. The equation of an ellipse has a simple form when the major and minor axes are chosen for the x-axis and y-axis.