An Elementary Course in Synthetic Projective Geometry Part 16

_A parabola has one and only one focus in the finite part of the plane._

[Figure 44]

FIG. 44

*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

[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. Princ.i.p.al 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 _princ.i.p.al 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 princ.i.p.al 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 pa.s.s 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_.

[Figure 48]

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 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 pa.s.s 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 princ.i.p.al axis of the conics in problems 9 and 10.