SL the semi-latus rectum or ordinate through S will = SX=2SA, which is the relation by which S was previously determined.

10. Properties of the secant and tangent. Art. 6 of Chap. IV. applies verbatim to the Parabola.

11. The tangent at any point makes equal angles with the focal distance of the point and the axis of the curve.

Let the tangent at P meet the directrix in F and the axis in T: join SP, SF, and draw the perpendicular PM. By the last Art. PSF is a right angle, also SP=PM; hence the right-angled triangles PSF, PMF having equal heights and hypotenuse common are equal in all respects: the angle SPF=MPF = PTS.

COR. 1. Hence SPST, and the perpendicular on the tangent from S will bisect the angle PST.

COR. 2. The angle between the focal distance and the tangent at any point cannot be a right angle except at the vertex, when SP, PM are in the same straight line and the tangent makes equal angles with them.

12. Tangents at the extremities of a focal chord inter

sect on the directrix in a right angle.

Let PF, QF, tangents at the extremities of the focal chord PSQ, intersect on the directrix in the point F and cut the axis in points T, T.

Then the angle PFQ

13. The foot of the perpendicular from the focus on any tangent, lies on the tangent at the vertex.

Let PR the tangent at P intersect SM in R. Then in the triangles SPR, MPR, SP=PM, and the angle SPR= MPR, hence the triangles are equal in all respects and the angle SRP = MRP, SM is at right angles to PR. Join AR; then SX is bisected in A and

SM in R:

AR is parallel to XM and is

perpendicular to the axis and touches

M

X

R

the parabola at A the vertex. Hence the foot of the perpendicular on PR from S lies on the tangent at A.

COR. If O be the middle point of SP, RO will be parallel to MP, and therefore perpendicular to AR, and the circle on SP as diameter will touch AR at R. AR is the limiting form of the auxiliary circle in the ellipse and 7

J. C. S.

hyperbola when the centre removes to a continually increasing distance.

14. ST: QN AS: AX, as in Chap. IV. § 11.

::

15. The tangents to a parabola from an external point subtend equal angles at the focus, as in Chap. IV. § 12.

16. The angle between two tangents to a parabola equals that subtended by either tangent at the focus.

Let SY, SZ be the perpendiculars from the focus on the tangents to the parabola at P and Q which intersect in O. Join SP, SQ, SO. Then the angle

ASY half ASP, and ASZ = half

ASQ.

=

Also the angle between the tangents equals the angle between the perpendiculars on them

=ZSY-ASZ-ASY

= ASQ ASP=†PSQ

=

OSP or OSQ.

17. The triangles SOP, SQO are similar and SO2 = SP. SQ.

The same construction being made as in the last proposition, we shall have the sum of the angles SOP, POZ= SOZ the sum of the interior and opposite angles of the triangle SQO the sum of SQO, OSQ.

=

=

And the angle PSOOSQ. Hence the triangles SOP, SQO are similar and SP: SO :: SO: SQ or SO2 = SP. SQ.

COR. If SY be the perpendicular from S on the tangent at P, YA will be the tangent at A, and SY2 = SA . SP.

18. The circle circumscribing the triangle formed by three tangents to a parabola passes through the focus.

Let two of the tangents intersect

in 0 and the third tangent cut them in P, Q and touch the curve in R. Join SR and produce QO to Y.

Then the angle OPQ = PSR,

and the angle

hence the angle

=PSR+ QSR = QSP:

A

the sum of the opposite angles POQ, PSQ of the quadrilateral OPSQ = the sum of POQ, POY= two right angles, and the quadrilateral can be inscribed in a circle. In other words, the circle circumscribing the triangle POQ will pass through S.

19. The tangents at the extremities of a chord intersect on the diameter of the chord. See Chap. v. § 7.

Let QT be the tangent at the point Q and QV the ordinate at that point to the diameter PV: to shew that PT=PV. For QV will be parallel to PR the tangent at P. Draw PO parallel to QT to meet QV in O. Join RO: being

the diagonal of the parallelogram POQR it will bisect the chord PQ which is the other diagonal.

Hence RO bisecting the chord PQ and passing through the intersection of the tangents at the extremities of the chord will be parallel to the axis of the curve and therefore to PV: RPVO, RTPO are parallelograms and PT=RO = PV.

=

COR. 1. If PN be the ordinate from any point P to the axis of the curve, and PT the tangent at that point, we shall have AT= AN.

COR. 2. To draw the tangents to the parabola from an external point T, draw TPV parallel to the axis meeting the curve at P; make PV equal to PT, and through V draw the chord QVq parallel to the tangent at P: QT, qT will be the tangents required.

The proposition of this Article results from the corresponding CV. CT = CP2 of the ellipse and hyperbola in the same way that AS AX results from the relation CS. CX

=

21. QV2=4 SP.PV.

We have already seen that if QR be the perpendicular distance of the extremity of a chord QVq from its diameter PV, QR 4 AS. PV.

=

To find the relation between QR and QV draw SY the perpendicular on the tangent at P and join AY which will be the tangent at A. Join Y SP. Then the angle SPYRPY= the interior and opposite angle RVQ, since QV is parallel to PY; the right- A

angled triangles SPY, QVR are similar and QV2: QR2 :: SP2: SY2. But SY AS. SP;

=