Let the hyperbola and its asymptotes be referred to the same system of conjugate diameters CX, CY, of which CX is parallel to Rr then CX bisects the ordinate to the asymptotes as well as that to the curve; that is, MR=Mr, and MP Mp; . PR=pr. 176. COR. 1. Let Rr move parallel to itself till the two points P, p coalesce, then, Rr becomes a tangent to the hyperbola, and therefore, by the property above proved, is bisected at the point of contact. COR. 2. Hence, PR. Pr=CD'. For PR. Pr= (MR-MP) (MR+MP) 177. PROP. 3. To find the equation to the hyperbola, when referred to its asymptotes (fig. 69.). Let P be any point in the hyperbola, PM, MC its rectangular co-ordinates, PR, RC its co-ordinates referred to the asymptotes CZ, Cz. Through R and P draw Rm, PQ parallel to PM, CX respectively. Let CM=x, MP=y, and angle RCX=0. x and The aim of the problem is to express and y' in terms of y, since the equation sought will then be obtained by substituting these values of x and y', in the equation Now y=PM=Rm- RQ=CR sin RCX-PR sin RCX; Hence, suppressing the accents, and denoting (a+b2) by m2, the equation to the hyperbola referred to its asymptotes, is xy = m2. 178. COR. 1. If Pp be drawn parallel to CR, then, the area of the parallelogram Rp is constant. For its area = RC. Cp.sin RCz =xy sin x, y = m2 sin x, y, whence the truth of the corollary. 179. COR. 2. Let the axes be parallel to the asymptotes, and originate at a point (a, ẞ). Then x=a+x', y=B+y Substituting these values of x and y in the equation xy=m3, we have x'y' +ay' +ßx'+aß — m2 = 0. Comparing this with the equation bxy + dy + ex +ƒ = 0, it appears that the general equation, when deprived of the terms involving y and x, represents an hyperbola, referred to axes which are parallel to the asymptotes. 180. PROP. 4. To find the equation to the tangent, the asymptotes being the axes. U The equation to a secant drawn through any two points (x', y') and (x", y") is .'. subtracting and adding x"y, we have x"y" — x'y' - x'y' + x'y' = 0, Suppose now, that the points (x, y), and (x", y') coincide; then the secant becomes a tangent, and x"=x′, y′′=y′; therefore the equation to the tangent is This equation, when reduced, assumes the very simple form xy+yx'=2m2............ (3). 181. COR. If a tangent applied at any point be produced to meet the asymptotes, the area of the triangle thus cut off is constant, (fig. 64.) For in the equation xy' +yx'=2m2 which is a constant quantity; whence the truth of the Cor. 182. PROP. 5. The asymptotes, and one point in the curve being given, to find the direction and magnitude of the principal diameters. (1) To find their direction. Bisect the angle, and the supplement of the angle, contained by the asymptotes; then the direction of the principal diameters is known. (2) To find their magnitude. The co-ordinates of the given point being x', y', we have .. the position and magnitude of the principal diameters are determined. 183. The principal properties of the ellipse, hyperbola, aud parabola, having now been deduced from their respective equations, we propose in the next place, to shew how such properties as are common to the three curves, may be investigated by means of their general equation. Previously, however, to entering upon this inquiry, it may be useful to bring under review the various equations, by which lines of the second order are characterized. These equations may be arranged as follows: I. Algebraical equations. (1) Let the curve be referred to its centre and principal diameter, then (2) Let the curve be referred to the principal diameter, and the tangent at its vertex, then 2 The general equation is y2= mx + nx2 which represents an ellipse, an hyperbola, a parabola, or a circle, according as n is negative, positive, 0, or= − 1. When the curve is referred to any system of conjugate diameters, the equations will be of the same form, and may be obtained from those above given, by merely changing a and b into a and b'. |