154. Focal properties of conics. We proceed to develop some theorems which will exhibit the importance of these points in the theory of the conic section. Draw a tangent to the conic, and also the normal at the point of contact P. These two lines are clearly conjugate normals. The two points T and N, therefore, where they meet the axis which contains the foci, are corresponding points in the involution considered above, and are therefore harmonic conjugates with respect to the foci (Fig. 44); and if we join them to the point P, we shall obtain four harmonic lines. But two of them are at right angles to each other, and so the others make equal angles with them (Problem 4, Chapter II). Therefore
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
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.