meet the parabola in another point; and the line between its points of section with the curve will be bisected by RH.

From Q, draw QS at right angles to the directrix; and from the centre Q, with the distance QS, describe a circle; this circle will evidently pass through the focus F (Art. 86), and touch the directrix in S (cor. 16.3). Join FR, and let QV cut FR in Y; then, since HK is at right angles to FR, and also bisects it (Art. 89, cor. 2); the angles at Y are right angles; and the point X, where the circle cuts FY the second time, lies between F and R. Also, FY YX (3.3). Make RT RS; draw TP parallel to AM, meeting QV in P; and through the points F, T, X, describe a circle. Then, since FR cuts the circle FXS, and RS touches it, FR.RX = RS? (36.3) = RT2; consequently, RT touches the circle FTX in

=

=

T (37.3). Hence the centre of that circle is in TP, which is at right angles to RT (19.3); it is also in VY, which bisects FX at right angles (cor. 1.3); it is therefore in P; the point P is of course in the parabola (Art. 86).

Now, PT, HR and QS being parallel, and TR = RS, it follows (2.6) that PV = VQ

Q. E. D.

Def. 11. Any line OR, parallel to the axis, is called a diameter of the parabola; the point H, where the diameter meets the curve, is called its vertex; 4HR is called the latus rectum of that diameter; the line PV or VQ, parallel to the tangent HK, is called an ordinate and HV an abscissa to the diameter он.

ART. 94. Let PZ be drawn at right angles to the diameter OR, and PV parallel to the tangent HK; then PZ2=4AF.VH.

Retaining the construction of the last article, let the tangent HK meet FR in a, and the diameter TP in c; draw cn parallel to FR, and join Aa. Then, since FR is bisected in a (Art. 89, cor. 2); and FX in Y (3.3); Ya or nc = XR: also, Pc VH (34.1). But (as was proved in Art. 92), the triangle FAa is similar to FaK or Pcn. Consequently,

=

As Fa: FA :: Pc : cn;

whence FA.Pc (or FA.VH) = Fa.nc (16.6). Hence, 4FA.VH = FR.RX = TR2 = PZ2

Q. E. D.

ART. 95. If, from two points P, A, ordinates, PV, AO, be drawn to any diameter OR, the squares of those ordinates shall be to each other in the same ratio as their abscissas; that is,

As PV2: AO :: VH: OH.

Draw PZ, AW, at right angles to OR; then (Art. 94) PZ3= 4FA.VH; and AW 4FA.OH; consequently (1.6),

PZ

=

AW :: VH: OH.

But the triangles PVZ, AOW, being similar,

PV: AO :: PZ2: AW2 :: VH: OH.

Q. E. D.

ART. 96. The square of any ordinate is equal to the rectangle of its abscissa, and the latus rectum of the diameter.

Let PV be an ordinate to the diameter OH; from the vertex of the axis let AO be drawn to the same diameter, parallel to PV; from H, draw HM at right angles to the axis. Then, since AOHK is a parallelogram,

Now,

OH = AK = (Art. 90) AM = HW .. OW = 20H.

=

AO' AW' + OW (47.1)=4AF.OH + 40H (Art. 94, and cor. 2.8.2) = 4RW.OH + 4WH.OH = 4RH.OH.

But (Art. 95),

As AO2 : PV2 :: OH : VH :: (1.6) 4RH.OH: 4RH.VH. Hence, PV2 = 4RH.VH.

Q. E. D.

ART. 97. A double ordinate passing through the focus of a parabola, is equal to the latus rectum of the diameter to which that ordinate is applied.

Let F be the focus; VH a diameter; EG the directrix; HD a tangent to the parabola; PFO, the line through the focus parallel to DH. Join FH; then, PV being parallel to DH, the angle DHG = FVH; and DHF HFV (29.1). But DHG = DHF (Art. 89); therefore, FVH = HFV; and HV = FH (6.1)= HG (Art. 86). Now, PV VO (Art. 93), and PV2 = 4GH.HV (Art. 96) = 4HG'. Hence PV = 2HG (cor. 2.8.2). Therefore, PO = 4HG.

=

The case of the double ordinate applied to the axis, is proved in Art. 87.

Q. E. D.

called the focus of the ellipse; and the line HI, passing through either focus at right angles to AE, meeting the ellipse in H and I, is called the latus rectum of the ellipse.

ART. 98. The rectangle of the abscissas AF.FB, into which the axis AB is divided by the focus, is equal to the square of the semi-axis CE.

Since AB is bisected in E, and divided unequally in F; AF.FB + FE2 = AE2 (5.2) = FC2 = FE2 + CE2 (47.1); therefore, AF.FB = EC". Q. E. D. Cor. Hence, HF is a third proportional to AE, EC. For (Art. 84),

whence

As AE2: EC2 :: AF.FB (EC2) : FH2;

AE: EC :: EC : FH.

ART. 99. If from L, any point in the ellipse, a line LN be drawn at right angles to the second or minor axis CD; then, As EC2 EB2 :: CN.ND: NL2.

:

Draw LM at right angles to AB; then (Art. 84),

As EB EC2 :: AM.MB (EB2 — EM2): ML2 or EN2.

Therefore (19.5),

As EB EC :: EM2: EC2

EN2 or CN.ND (5.2);

M