angles which these tangents make with the tangent to the confocal ellipse passing through P will be constant while P moves on that ellipse. For if a and A be the semi-axes of the interior ellipses, we have, from what has been just proved, an expression not involving a", and therefore the same for every point on the ellipse a'. THE ASYMPTOTES. 195. We have hitherto discussed properties common to the ellipse and the hyperbola. There is, however, one class of properties of the hyperbola which have none corresponding to them in the ellipse, those, namely, depending on the asymptotes, which in the ellipse are imaginary. We saw that the equation of the asymptotes was always obtained by putting the terms containing the highest powers of the variables = 0, the centre being the origin. Thus the equation. of the curve, referred to any pair of conjugate diameters, being Hence the asymptotes are parallel to the diagonals of the parallelogram, whose adjacent sides are any pair of conjugate semidiameters. For, the equation of is parallel to the other(see Art.167). T Hence, given any two conjugate diameters, we can find the asymptotes; or, given the asymptotes, we can find the diameter conjugate to any given one; for if we draw 40 parallel to one asymptote, to meet the other, and produce it till OB= A0, we find B, the extremity of the conjugate diameter. 196. The portion of any tangent intercepted by the asymptotes is bisected at the curve, and is equal to the conjugate diameter. This appears at once from the last Article, where we have proved AT=b'AT'; or directly, taking for axes the diameter through the point and its conjugate, the equation of the asymptotes is Hence, if we take x=a', we have y=+b'; but the tangent at A being parallel to the conjugate diameter, this value of the ordinate is the intercept on the tangent. 197. If any line cut a hyperbola, the portions DE, FG, intercepted between the curve and its asymptotes, are equal. For, if we take for axes a diameter parallel to DG and its conjugate, it appears from the last Article that the portion DG is bisected by the diameter; so is also the portion EF; hence DE = FG. B M The lengths of these lines can immediately be found, for, from the equation of the asymptotes (2 – 3 =0), y (=DM=MG) = ± − x. Again, from the equation of the curve we have Hence and 12 = we have 198. From these equations it at once follows that the rectangle DE.DF is constant, and b". Hence, the greater DF is, the smaller will DE be. Now, the further from the centre we draw DF the greater will it be, and it is evident from the value given in the last article, that by taking x sufficiently large, we can make DF greater than any assigned quantity. Hence, the further from the centre we draw any line, the less will be the intercept between the curve and its asymptote, and by increasing the distance from the centre, we can make this intercept less than any assigned quantity. 199. If the asymptotes be taken for axes, the coefficients g and f of the general equation vanish, since the origin is the centre; and the coefficients a and b vanish, since the axes meet the curve at infinity (Art. 138, Ex. 4); hence the equation reduces to the form xy=k". The geometrical meaning of this equation evidently is, that the area of the parallelogram formed by the coordinates is constant. The equation being given in the form xy=k2, the equation of any chord is (Art. 86), or = Making xx" and y' =y", we find the equation of the tangent From this form it appears that the intercepts made on the asymptotes by any tangent = 2x and 2y'; their rectangle is, therefore, 4k2. Hence, the triangle which any tangent forms with the asymptotes has a constant area, and is equal to double the area of the parallelogram formed by the coordinates. Ex. 1. If two fixed points (x'y', x'y') on a hyperbola be joined to any variable point on the curve (x""y"), the portion which the joining lines intercept on either asymptote is constant. The equation of one of the joining lines being x""'y + y'x = y'x'" + k2, the intercept made by it from the origin on the axis of x is found, by making y = 0, to be x'"+x'. Similarly the intercept from the origin made by the other joining line is x""+x", and the difference between these two (x' - x") is independent of the position of the point x""y"". Ex. 2. Find the coordinates of the intersection of the tangents at x'y', x′′y′′. 200. To express the quantity k in terms of the lengths of the axes of the curve. Since the axis bisects the angle between the asymptotes, the coordinates of its vertex are found, by putting x=y in the equation xy= k*, to be x=y=k. Hence, if be the angle between the axis and the asymptote a = 2k cos 0, (since a is the base of an isosceles triangle whose sides = k and base angle = 0), but (Art. 165) And the equation of the curve, referred to its asymptotes, is a2 + b2 201. The perpendicular from the focus on the asymptote is equal to the conjugate semi-axis b. For it is CF sine, but CF= √(a2 + b2), and sin 0 = b √ (a2 + b2) This might also have been deduced as a particular case of the property, that the product of the perpendiculars from the foci on any tangent is constant, and b. For the asymptote may be considered as a tangent, whose point of contact is at an infinite distance (Art. 154), and the perpendiculars from the foci on it are evidently equal to each other, and on opposite sides of it. 202. The distance of the focus from any point on the curve is equal to the length of a line drawn through the point parallel to an asymptote to meet the directrix. са For the distance from the focus is e times the distance from the directrix (Art. 186), and the distance from the directrix is to the length of the parallel line as cos 1 = Art. 167 is to 1. e 7) Hence has been derived a method of describing the hyperbola by continued motion. A ruler ABR, bent D B A D' 'F R P |