(13.) The parabola referred to any diameter. A diameter and the tangent at its vertex being the axes, the equation is

y' = 4m,.,

or QV2=4 SP.PV.
The parameter=4SP.
2m.gF=PF.FP.

PF 2m.PV.

If 4m, 4m, are the parameters corresponding to the ordinates Pp, Qq respectively, then

4m. EM=PM.Mp.

PM.Mp : QM.Mq:: m : mę.
PM : Mp :: DE : EM.
The subtangent NT is bisected by the curve at A.

If from the several points of any line given in position pairs of tangents be drawn to a parabola, the chords-joining the corresponding points of contact will pass through the same point. The chord of curvature through the focus=4 SP.

2 Spi The radius of curvature =

SA. (H. C. S. 71-84; Hust. Prop. 9.--15; Biot, 191-223;

G. G. A. Ch. xii.)

THE ELLIPSE.

(14.) The Ellipse referred to its awes. Let S be the focus, and

PQ perpendicular to EQ the directrix; then

SP : PQ :: e : 1; e being a constant quantity, and < 1,

The equation referred to the axis, and a tangent at the vertex is

72

gø= (2ax– xo),
AN.Na : NP :: AC : BC.
The equation referred to the centre and axes is

AS. Sa= BC = AC. SL.
The intersection of an ellipse with the straight line,

y=ax +ß, may be determined from the equation

26*2 b?(B– aRa*) ya

=0;

aʻa? +62 which has only two roots, hence a straight line cannot cut an ellipse in more than two points.

aľa? + 72Y+

The equation to a tangent at the point (21,9,) is

yy
+ = 1.

a? 62
Let PT be a tangent at the point P, then

CN.CT= AC; Cn.Ct + BC
PF.PG=BC

The subtangent, NT=

2 1 If the ordinate NP be produced to meet the circumference of a circle described on Aa in R, the tangents at the points P and R will meet the axis produced in the same point. The equation to the normal PG is

a

Yi
Y - Y1

(x — ,).

BC
The subnormal NG= CN.

AC2

To draw a tangent to an ellipse from a given point (2392) without it: the point of contact (x,y) must be determined from the equations

yey 2.X

=1, a?

+

2

YA

+ =1.

62 a? (H. C. S. 89—111; Hust. Prop. 8–12; Biot, 133_60.) (15.) The ellipse referred to the focus. Let S, H, be the foci, then SP-a-ex,

HP=a +ex.

SP + HP=2a= Aa.
The polar equation to the ellipse, the focus being the pole, is

1-e?
1+ e cos e

U=a

BC2
SP=

AC – SC. cos PSN

1 1 2

+
SP Sp SL
If the centre is the pole, the equation is

1-e
U=a.

1-(e cos 0) The angle SPT=HPt.

If SY, HZ be drawn perpendicular to the tangent, the points Y, Z are always in the circumference of a circle described on Aa.

PI=AC.

SY.HZ=BC. (H. C. S. 112--22; Hust. Prop. 1–7.) (16.) The ellipse referred to any conjugate diameters.

If CD is conjugate to CP, conversely CP is conjugate to CD.

A tangent at P is parallel to CD.

The axes are the only conjugate diameters that are perpendicular to each other.

The equation to an ellipse referred to the centre and any conjugate diameters a,, b, is

y

=1. be

a

2

If the extremity of the diameter is the origin, then

b?
= 2a)

ai
PV.Vp : QV :: CP : CD?.
CV.CT=CP%; and Cu.Ct=CD?.

If any chords Qq, Rr, parallel to CD, CP, respectively, intersect each other in 0, then

Q0.09 : RO.Or :: CD : CP?.
If a, b, are any conjugate diameters,

az? +b = a + b?.
The area of all circumscribing parallelograms is constant, and

= 4CD. PF-4 AC. BC.
If a;=672
then a, =} (a? +62)];

2 ab
a +62
b

sin a ,b,=

tan aga =

SP.HP=PT. Pt=CD.

(See Fig. p. 163.) AM .am=

= BC? Let Pp be the diameter drawn through any point P, whose co-ordinates are & q, Yı, and PQ, Qp the supplemental chords : the equation to PQ being

y-yı=a (x – 0,), the equation to Qp is

b b,
y+y= (8 + x).

a 2a If the chords be drawn from the extremities of the axis major, their equations are

y=a (v + a);

6,2
y= x

2

2

aa