If two diameters be drawn parallel to any two supplemental chords, they are conjugate to each other. tan APa= 2ab2 (H. C. S. 127-64; Hust. Prop. 13-8; Biot, 161-90.) (17.) The Hyperbola referred to its axes. If the curve is referred to the axis, and a tangent at the vertex, the equation is b2 (2 ax + x2). a2 If the origin is at the centre, the equation is The lines in the above figure are analogous to those in the ellipse, on which the same letters are placed. The intersections of the hyperbola with the straight line, 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 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 (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 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 In order that the diameter may meet the curve, a must If the diameter a, is conjugate to b1, b1 is conjugate to a 1 The equation to the diameter passing through (x1,y1) is The equation to an hyperbola referred to the centre, and two conjugate diameters a1, b1, is, If P, the extremity of a,, 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. 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. 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 a2 If the asymptotes be the axes, and the centre the origin, the equation to the hyperbola is xy= 1 (a2 + b2). |