BOOK IX.

CONIC SECTIONS.

THE PARABOLA.

850. DEF. A parabola is a curve which is the locus of a point that moves in a plane so that its distance from a fixed point in the plane is always equal to its distance from a fixed line in the plane.

851. DEF. The fixed point is called the focus; and the fixed line, the directrix.

852. A parabola may be described by the continuous motion. of a point, as follows:

Place a ruler so that one of its edges shall coincide with the directrix DE. Then place a right triangle with its base edge in contact with the edge of the ruler. Fasten one end of a string, whose length is equal to the other edge BC, to the point B, and the other end to a pin fixed at the focus F. Then slide the triangle BCE along the directrix, keeping the string tightly pressed against the ruler by the point of a pencil P. The point P will describe a parabola; for during the motion we always have PF equal to PC.

· PROPOSITION I. PROBLEM.

853. To construct a parabola by points, having given its focus and its directrix.

Let F be the focus, and CDE the directrix.

Draw FD to CE, meeting CE at D. Bisect FD at A.

Then A is a point of the curve.

$ 850 Through any point M in the line DF, to the right of A, draw a line to CE.

With F as centre and DM as radius, draw arcs cutting this line at the points P and Q.

Then P and Q are points of the curve.

=

Ax. 1

§ 850

DM = PF = QF.

=

.. PC PF, and QE QF.

Therefore, P and Q are points of the curve.

In this way any number of points may be found; and a continuous curve drawn through the points thus determined is the parabola whose focus is F and directrix CDE.

Q. E. F.

854. DEF. The point A is called the vertex of the curve. The line DF produced indefinitely in both directions is called the axis of the curve.

855. DEF. The line FP, joining the focus to any point P on the curve, is called the focal radius of P.

856. DEF. The distance AM is called the abscissa, and the distance PM the ordinate, of the point P.

857. DEF. The double ordinate LR, through the focus, is called the latus rectum or parameter.

858. COR. 1. The parabola is symmetrical with respect to its axis.

For FPFQ (Const.), and, therefore, PM = QM.

§ 210

§ 149

of the

859. COR. 2. The curve lies entirely on one side perpendicular to the axis erected at the vertex; namely, on the same side as the focus.

For any point on the other side of this perpendicular is obviously nearer to the directrix than to the focus.

860. COR. 3. The parabola is not a closed curve.

For any point on the axis of the curve to the right of F is evidently nearer to the focus than to the directrix. Hence, the parabola QAP cannot cross the axis to the right of F.

861. COR. 4. The latus rectum is equal to 4 AF.

NOTE. In the following propositions, the focus will be denoted by F, the vertex by A, and the point where the axis meets the directrix by D.

PROPOSITION II. THEOREM.

862. The ordinate of any point of a parabola is the mean proportional between the latus rectum and the abscissa.

Let P be any point, AM its abscissa, PM its ordinate.

To prove that PM2 = 4 AF × AM.

863. COR. 1. The greater the abscissa of a point, the greater the ordinate.

864. COR. 2. The squares of any two ordinates are as the abscissas.

PROPOSITION III. THEOREM.

865. Every point within the parabola is nearer to the focus than to the directrix; and every point without the parabola is farther from the focus than from the directrix.

Proof. In the AQPF, QF < QP + PF.

.. QF <QP + PC.

.. QF <QC.

§ 138

2. Let Q' be a point without the curve.

To

prove that

Q'F> Q'C.

Draw Q'F.

§ 138

Q. E. D.

Proof. In the AQ'FP, Q'F> PF - PQ'.

866. COR. A point is within or without a parabola according as its distance from the focus is less than, or greater than, its distance from the directrix.

867. DEF. A straight line which touches, but does not cut, a parabola, is called a tangent to the parabola. The point where it touches the parabola is called the point of contact.