159. Eccentricity. By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the parabola. This ratio is called the eccentricity.
Fig. 48
160. Sum or difference of focal distances. The ellipse and the hyperbola have two foci and two directrices. The eccentricity, of course, is the same for one focus as for the other, since the curve is symmetrical with respect to both. If the distances from [pg 96] a point on a conic to the two foci are r and r', and the distances from the same point to the corresponding directrices are d and d' (Fig. 47), we have r : d = r' : d'; (r ± r') : (d ± d'). In the ellipse (d + d') is constant, being the distance between the directrices. In the hyperbola this distance is (d - d'). It follows (Fig. 48) that
In the ellipse the sum of the focal distances of any point on the curve is constant, and in the hyperbola the difference between the focal distances is constant.
PROBLEMS
1. Construct the axis of a parabola, given four tangents.
2. Given two conjugate lines at right angles to each other, and let them meet the axis which has no foci on it in the points A and B. The circle on AB as diameter will pass through the foci of the conic.