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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)
2 CD Chord of curvature through C=
2 CD Chord through the focus=
(Hust. Prop. 19.) AC
THE HYPERBOLA. (17.) The Hyperbola referred to its axes.
SP : PQ :: e: 1; e being > 1.*
If the curve is referred to the axis, and a tangent at the vertex, the equation is
(2ax + 20%).
(202 — a);
* 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?.
y= ax + ß
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
62 BC2 The subnormal, NG=
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
= a” 62
62 (H.C. S. 172—94; Hust. Prop. 7, 8; Biot, 224_44.)
(18.) The hyperbola referred to the focus.
SP=ex — a; and HP=ex + a.
1+e cos If H be the pole, then
AC - ŚC.cos PSN
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.
(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
(6o – aʼa)!
In order that the diameter may meet the curve, a must b
If the diameter a, is conjugate to b, b, is conjugate to an.
X 1 the equation to the conjugate diameter is
ar y The equation to an hyperbola referred to the centre, and two conjugate diameters an, bq, is,
= 1. b,
If P, the extremity of an, is the origin, the equation is
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?.
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.
y-y= a (x - 2), the equation to mp is
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 =
If the asymptotes be the axes, and the centre the origin, the equation to the hyperbola is
xy=* (a + b).