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 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.
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.
3. Given the axes of a conic in position, and also a tangent with its point of contact, to construct the foci and determine the length of the axes.
4. Given the tangent at the vertex of a parabola, and two other tangents, to find the focus.
5. The locus of the center of a circle touching two given circles is a conic with the centers of the given circles for its foci.
6. Given the axis of a parabola and a tangent, with its point of contact, to find the focus.
7. The locus of the center of a circle which touches a given line and a given circle consists of two parabolas.
8. Let F and F’ be the foci of an ellipse, and P any point on it. Produce PF to G, making PG equal to PF’. Find the locus of G.
9. If the points G of a circle be folded over upon a point F, the creases will all be tangent to a conic. If F is within the circle, the conic will be an ellipse; if F is without the circle, the conic will be a hyperbola.
10. If the points G in the last example be taken on a straight line, the locus is a parabola.
11. Find the foci and the length of the principal axis of the conics in problems 9 and 10.
12. In problem 10 a correspondence is set up between straight lines and parabolas. As there is a fourfold infinity of parabolas in the plane, and only a twofold infinity of straight lines, there must be some restriction on the parabolas obtained by this method. Find and explain this restriction.
13. State and explain the similar problem for problem 9.