CHAPTER II.

TANGENT-PROPERTIES OF THE PARABOLA.

16. IF the extremities of a chord be made to approach one another continuously the chord tends to assume a certain limiting position; and in this limiting position, viz. when its extremities have become coincident, the chord produced indefinitely has become a Tangent. Thus in fig. p. 14 suppose to move along the curve up to Q. Then ultimately QQ' becomes the tangent at Q.

There are various ways in which a chord may be made to assume a position of tangency. It is often convenient to use the following.

17. In fig. p. 14 Qq is a chord bisected by the diameter TV. Suppose this chord to move parallel to itself into a new position 'q'. It is still bisected by TV, say in V'.

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Suppose it now to move continuously in the same way until its middle point reaches the end of the diameter TV; then each of the portions QV, qV, which have been diminishing continuously, will have vanished, and the two points in which the chord met the curve will have been brought into coincidence; that is to say the chord will have become a tangent, viz. at the end P of the diameter which bisects Qq. Hence

The tangent at the extremity of any diameter is parallel to the ordinates of that diameter.

In particular

The tangent at the vertex is parallel to the principal ordinates, or perpendicular to the axis.

18. To take another example:-In Art. 14 we have shewn that the rectangles contained by the segments of any two intersecting chords are to one another as the lengths of the parallel focal chords. This holds whether the point of intersection be internal or external to the curve. Let it be external as T in fig. p. 14.

Then TQ. TQ is to Tq. Tq in the ratio of the focal chords to which QQ', qq' are parallel.

Let QQ move parallel to itself until it becomes the tangent at some point Q1. Then TQ. TQ' becomes TQ12.

2

In like manner let qq' become the tangent at some point 4. Then Tq. Tq' becomes Tq.

2

Therefore the squares of any two tangents TQ,, Tq, are to one another in the ratio of the focal chords to which they are parallel.

19. Again (9), if a diameter meet the directrix in O, then every chord QQ bisected by that diameter is perpendicular to OS. Therefore in the limit the tangent at P (fig. p. 7) the extremity of the diameter OV is perpendicular to OS, or, in fig. p. 16, to MS. And conversely, if the focal perpendicular on the tangent at P meets the directrix in M, then PM is a diameter, and is therefore parallel to the axis or perpendicular to the directrix.

20. We might also, after the method of Euclid, regard a tangent as a straight line which meets a conic and being produced does not cut it, or which, except at the point of contact, lies wholly on the convex or outer side of the conic. We shall briefly indicate the method to be pursued when the Euclid definition is adopted.

From any point P on the parabola (fig. p. 16) let fall perpendiculars PM, PY on the directrix and SM respectively. Then will PY be the tangent at P. For since evidently Y is the centre of SM, therefore if t be any point other than P on PY, then St TM. Therefore St is greater than the perpendicular distance of t from the directrix. Hence it may be shewn that every point t on PY lies on the convex side of the parabola, or PŸ is the tangent at P.

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21. PROP. VI. Tangents at the extremities of any chord intersect on the diameter which bisects the chord.

Let Q, Q' and q, q' be adjacent extremities of any two parallel chords, and let QQ, qq' meet in T, then shall Tlie on the diameter which bisects the chords.

For let TV, drawn to the centre of Qq, cut Q'q' in V'. Then by parallels,

Q'V' : q'V' =QV : qV.

But QV-qV. Therefore Q'V'=qV'.

Hence TV, since it bisects both Qq and Q'q', is the diameter which bisects chords parallel to Qq. [8.

Now as Q'q' moves parallel to itself up to Qq, the point of intersection Talways lies on the diameter through V. And this being true always, is true in the limit, viz. when QQ, qq' become the tangents at Q, q.

Hence the tangents at the ends of the chord Qq meet on the diameter which bisects the chord.

22. Conversely, if the tangents at P, Q meet in R, the straight line drawn through R to bisect the chord of contact PQ is a diameter.

23. PROP. VII. If PV be the abscissa of any point Qon the parabola, and if the tangent at Q meet the diameter PV in T, then

Let the tangent at P, which is parallel to the ordinate QV, meet QT in R. Complete the parallelogram QRPO, viz. by drawing PO parallel to RQ. Then the diagonal RO bisects the diagonal PQ. That is to say, RO bisects the chord of contact of the tangents RP, RQ. Therefore RO is a diameter (22), and is parallel to any other diameter, as PV, since all diameters are parallel to the axis.

Hence by parallels,

PV=RO=PT.

24. The following particular case is to be noticed.

If PN be the principal ordinate of P, then ANAT.

Draw PM perpendicular to the directrix, and let SM meet the tangent at P in Y.

Then in the triangles SYP, MYP, since

SP, PY = MP, PY,

each to each, and the angles at Y are right angles,

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26. COR. If the tangent meet the directrix in R, it readily follows that RSP = RMP = a right angle. But this is best proved independently as in Art. 105.

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27. COR. If the tangent meet the axis in T, then by parallels and by the proposition, ‹ STP = MPT = SPT.

28. PROP. IX. If from any point T on the tangent at P perpendiculars TL, TN be let fall on SP and the directrix,

then

If the tangent meet the directrix in R the angle RSP is

a right angle.

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