The lines joining a point on the conic to the foci make equal angles with the tangent.

It follows that rays from a source of light at one focus are reflected by an ellipse to the other.

155. In the case of the parabola, where one of the foci must be considered to be at infinity in the direction of the diameter, we have

[Figure 45]

FIG. 45

A diameter makes the same angle with the tangent at its extremity as that tangent does with the line from its point of contact to the focus (Fig. 45).

156. This last theorem is the basis for the construction of the parabolic reflector.  A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector.

157.  Directrix.  Principal axis.  Vertex. The polar of the focus with respect to the conic is called the directrix.  The axis which contains the foci is called the principal axis, and the intersection of the axis with the curve is called the vertex of the curve.  The directrix is at right angles to the principal axis.  In a parabola the vertex is equally distant from the focus and the directrix, these three points and the point at infinity on the axis being four harmonic points.  In the ellipse the vertex is nearer to the focus than it is to the directrix, for the same reason, and in the hyperbola it is farther from the focus than it is from the directrix.

[Figure 46]

FIG. 46

158.  Another definition of a conic. Let P be any point on the directrix through which a line is drawn meeting the conic in the points A and B (Fig. 46).  Let the tangents at A and B meet in T, and call the focus F.  Then TF and PF are conjugate lines, and as they pass through a focus they must be at right angles to each other.  Let TF meet AB in C.  Then P, A, C, B are four harmonic points.  Project these four points parallel to TF upon the directrix, and we then get the four harmonic points P, M, Q, N.  Since, now, TFP is a right angle, the angles MFQ and NFQ are equal, as well as the angles AFC and BFC.  Therefore the triangles MAF and NFB are similar, and FA :  FM = FB :  BN.  Dropping perpendiculars AA and BB’ upon the directrix, this becomes FA :  AA’ = FB :  BB’.  We have thus the property often taken as the definition of a conic: 

The ratio of the distances from a point on the conic to the focus and the directrix is constant.

[Figure 47]

FIG. 47

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.