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If two diameters be drawn parallel to any two supplemental chords, they are conjugate to each other.

2 ab tan APa=

y (a? 62)
(H. C. S. 127–64; Hust. Prop. 13–8; Biot, 161-90.)

CD2
Radius of curvature =

PF

2 CD Chord of curvature through C=

CP

2 CD Chord through the focus=

(Hust. Prop. 19.) AC

THE HYPERBOLA. (17.) The Hyperbola referred to its axes.

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yo =

SP : PQ :: e: 1; e being > 1.*

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

72

(2ax + 20%).

a
AN. Na : NP2 :: AC2 : BC2.
If the origin is at the centre, the equation is
62

x2

ya
ye:

(202 a);
a); or =1.

a

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* The lines in the above figure are analogous to those in the ellipse, on which the same letters are placed.

The equation to the equilateral hyperbola is

20? yo=a?.
72 BC2
SL +

AC
The intersections of the hyperbola with the straight line,

y= ax + ß
may be found by the solution of the equation

26ß 6° (do ao-3°) y+

0 aʻa’ - 62 a'a' - 6 which has only two roots; a straight line therefore cannot cut an hyperbola in more than two points.

The equation to a tangent at the point (Q1,4.) is

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62 BC2 The subnormal, NG=

CN. az".

AC To draw a tangent from a point (x,y) without the hyperbola: the point (avy.) at which the tangent meets the curve must be determined from the equations

XX1 y.

= a” 62

-1,

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yi?

=1.

62 (H.C. S. 172—94; Hust. Prop. 7, 8; Biot, 224_44.)

U= a

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a
1

(18.) The hyperbola referred to the focus.

SP=ex a; and HP=ex + a.
HP- SP=2 AC.
The polar equation, the focus S being the pole, is

e-1

1+e cos If H be the pole, then

e-1

e cos
BC
SP=

AC - ŚC.cos PSN
1 1 2

+ +
SP Sp SL
SP.Sp=ļa (e* – 1)(SP+ Sp).
If the centre is the pole, the equation is

e-1

1- (e cos ) LHPT=_SPT.

If SY and HZ be drawn perpendicular to PT, the locus of the points Y and Z is a circle described on Aa.

PI=Pi= AC.
SY.HZ=BC2.

(H.C. S. 195–205; Biot, 259—63; Hust. Prop. 1–6.) (19.) The hyperbola referred to any system of conjugate diameters.

The locus of the points of bisection of all parallel chords is a diameter. The co-ordinates of the points of intersection of

any diameter y=ax with the curve are ab

aba
x = +
(62 - aʼa”)??

y=+

(6o aʼa)!

U=a.

2

Y

In order that the diameter may meet the curve, a must b

b

be <

and >

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a

If the diameter a, is conjugate to b, b, is conjugate to an.
The equation to the diameter passing through (@134.) is

Y1
y=

X 1 the equation to the conjugate diameter is

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a

y =

ar y The equation to an hyperbola referred to the centre, and two conjugate diameters an, bq, is,

X2

= 1. b,

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2

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If P, the extremity of an, is the origin, the equation is

,
=

an
PV.Vp : QV :: CP2 : CD.
CV.CT=CP2; Cv.ct=CD?.

If from the several points of a straight line given in position, pairs of tangents be drawn to an hyperbola, the lines joining the corresponding points of contact will all pass through the same point.

Q0.09 : RO.Or :: CD? : CP?.
a,. -6°=a? 6%.

If tangents to the conjugate hyperbolas be drawn at the extremities of the conjugate diameters, the area of the parallelogram is constant, and =4CD.PF=4 AC. BC.

SP.HP=CD?.
PT.Pt=CD.
If the equation to PM is

y-y= a (x - 2), the equation to mp is

6,
y+y= (x + x).

a'a If diameters be drawn parallel to any two supplemental chords, they are conjugate.

(H.C. S. 206—45; Biot, 245—58; Hust. Prop. 15—8.) (20.) The asymptotes of the hyperbola. If the origin is at the centre, and a, b are any system of conjugate diameters, the equation to the asymptotes is

62 yo =

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RQ=rq
QI.Qi=BC%.
QR.Qr=CD?.

If the asymptotes be the axes, and the centre the origin, the equation to the hyperbola is

xy=* (a + b).

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