Basic Technical Mathematics, 11th Edition

21.5 The Ellipse

• Foci • Vertices • Major Axis • Minor Axis • Standard Equation – Axis along x-axis • Standard Equation – Axis along y-axis • Calculator Display

The next important curve is the ellipse. An ellipse is defined as the locus of a point P(x, y) that moves so that the sum of its distances from two fixed points is constant. These fixed points are the foci of the ellipse. Letting this sum of distances be 2a and the foci be the points $(\hspace{0.17em}-\hspace{0.17em}c\hspace{0.17em},\hspace{0.17em}0)$ and (c, 0), we have

$$\sqrt{{(x\hspace{0.17em}-\hspace{0.17em}c)}^2\hspace{0.17em}+\hspace{0.17em}{y}^2}\hspace{0.17em}+\hspace{0.17em}\sqrt{{[\hspace{0.17em}x\hspace{0.17em}-\hspace{0.17em}(\hspace{0.17em}-\hspace{0.17em}c)]\hspace{0.17em}}^2\hspace{0.17em}+\hspace{0.17em}{y}^2}\hspace{0.17em}=\hspace{0.17em}2a$$

See Fig. 21.57. The ellipse has its center at the origin such that c is the length of the line segment from the center to a focus. We will also see that a has a special meaning. Now, from Section 14.4, we see that we should move ...