CHAPTER III.

THE PARABOLA.

Def. A PARABOLA is the curve traced out by a point which moves in such a way that its distance from a fixed point, called the Focus, is always equal to its perpendicular distance from a fixed straight line, called the Directrix.

In Prop. XII., p. 15, a Diameter of a conic has been defined as the straight line which bisects a system of parallel chords. In the parabola all diameters are parallel to the axis (Prop. 1.), and therefore to one another.

A diameter of a parabola is sometimes defined as a straight line parallel to the axis. In this case it may be shown, conversely, that every diameter bisects a system of parallel chords.

The focal chord parallel to the tangent at any point is said to be the Parameter of the diameter passing through that point.

The term ordinate is not confined to straight lines measured perpendicular to the axis of the parabola; but, if QV be drawn from any point Q on the curve, parallel to the tangent at P and meeting the diameter through P in V, then QV is said to be the Ordinate of the point Q, with reference to the diameter through P.

Also PV is called the Abscissa of Q.

Note. The terms ordinate and abscissa usually have reference to the axis. The context will determine when they are to be understood in their more general sense.

Several propositions that have been proved generally for conics assume simpler forms in the case of the parabola. Thus, in Prop. II., p. 7, since SA=AX, therefore SL=TN. Similarly SG=SP. [Prop. IX., p. 12.

Also PK, or the semi-latus rectum, is equal to SX or 2SA. [Prop. x., p. 12. PROP. I. The middle points of all parallel chords lie on a straight line parallel to the axis.

Let a straight line through the focus S meet the directrix

in M. Draw any chord Qq perpendicular to MS and meeting it in y. Let QN, qn be perpendiculars on the directrix. Then

Mn2 = Mq2 — qn2
=Mq" - Sq3.

[Euc. I., 47.

[Def.

But Mq' is equal to My+qy2, and Sq2 to Sy2+qy3. [Euc. I., 47.

By subtraction My" - Sy2 = Mq" — Sq2 = Mn2.

Similarly it may be shown that MN is equal to My2 - Sy3, and therefore to Mn2.

Hence M is the middle point of Nn.

Draw MO, parallel to the axis, to meet Qq in O.

Then O is the middle point of any chord Qq drawn perpendicular to SM. Therefore MO bisects all chords parallel to Qq.

PROP. II. The tangent at P makes equal angles with SP, PM, where PM is perpendicular to the directrix. Also, if the tangent meet the axis in T, then

L STP=SPT.

Let the tangent at P meet the directrix in R. Then PSR is a right angle. [Prop. I., p. 6.

Also, in the right-angled triangles

SPR, MPR, the sides SP, PM are

equal, and PR is common.

Hence the remaining angles are

equal, each to each, so that

L SPR=MPR.

Hence also the supplementary angles, which RP produced makes with SP, PM, are equal.

Produce PR to meet the axis in T.

M

R

N

Then ▲ SPT=MPT=alternate angle STP.

COR. The exterior angle PSO (fig., Prop. III.) is therefore equal to 2STP.

[Euc. I., 32. Note. Conversely, if the straight line YPR be drawn (fig., Prop. I.) making equal angles with SP, PM, this straight line will be the tangent at P.

For, if R be any point on the straight line, then, since SP, PR=MP, PR, each to each, and the included angles are equal, by construction; therefore SP is equal to PR, and consequently greater than the perpendicular distance of R from the directrix.

Hence it may be shown that all points on the straight line PR, except P, lie on the convex side of the curve, or that PR is the tangent at P.

PROP. III. The external angle between any two tangents is equal to the angle which either of them subtends at the focus. Let the tangents at P, Q, intersect in R, and meet

P

TA S

the axis in T, U, respectively. Take any point O in AS

Hence, the angles PSR, QSR being equal (Prop. VI., p. 9), either of them is equal to TRU.

PROP. IV. The subnormal is equal to 2AS or half the

latus rectum.

Let the tangent and normal at P meet the axis in T, G, respectively. Let PM be perpendicular to the directrix, and PN the ordinate of P. [fig., Prop. v.

Hence NG is equal to SX, that is, to 2SA, or to the

ordinate through S.

PROP. V. The subtangent is equal to twice the abscissa.

Let the tangent at P meet the axis in T.

Draw PM perpendicular to the directrix and let PN be

But SA=AX (Def.). Therefore AT=AN, or NT=2AN.

PROP. VI. If PN be the ordinate of P, then

PN 4AS.AN.

Draw the normal PG. Then TPG, PNG are right angles.

Or thus: since S divides AN so that AN + AS = XN, therefore

PROP. VII. The diameter through any point P, on the

curve, meets the ordinate of Qin V, and the tangent at Q

in T. To

To prove that

PV=PT.