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

tan APa=

2ab2
Y1 (a2 — b2)

(H. C. S. 127-64; Hust. Prop. 13-8; Biot, 161-90.)

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(17.) The Hyperbola referred to its axes.

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

b2

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(2 ax + x2). a2

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If the origin is at the centre, the equation is

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The lines in the above figure are analogous to those in the ellipse, on which the

same letters are placed.

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The intersections of the hyperbola with the straight line,

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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 (x,y) is

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To draw a tangent from a point (x,y) without the hyperbola: the point (x,y1) at which the tangent meets the curve must be determined from the equations

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(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.

HP SP=2AC.

The polar equation, the focus S being the pole, is

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

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(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

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

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If the diameter a, is conjugate to b1, b1 is conjugate to a

1

The equation to the diameter passing through (x1,y1) is

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The equation to an hyperbola referred to the centre, and two conjugate diameters a1, b1, is,

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

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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.

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

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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

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a2

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

xy= 1 (a2 + b2).

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