165. To investigate the figure of the hyperbola. The terms axis major and axis minor not being applicable to the hyperbola (Art. 160), we shall call the axis of x the transverse axis, and the axis of y the conjugate axis. Now b2 - (a+b) sin'0, the denominator in the value of p2, will plainly be greatest when = 0, therefore, in the same case, ρ will be least; or the transverse axis is the shortest line which can be drawn from the centre to the curve. As increases, p continually increases, until 2 ρ when the denominator of the value of p becomes = 0, and p becomes infinite. After this value of 0, p2 becomes negative, and the diameters cease to meet the curve in real points, until again when p again becomes infinite. It then decreases regularly as increases, until becomes = 180°, when it again receives its minimum value = a. The form of the hyperbola, therefore, is that represented by the dark curve on the figure, next article. L' K 166. We found that the axis of y does not meet the hyperbola in real points, since we obtained the equation y=-b" to determine its point of intersection with the curve. We shall, however, still mark off on the axis of y portions CB, CB'=±b, and we shall find that the length CB has an important connexion with the B A B curve, and may be conveniently called an axis of the curve. In like manner, if we obtained an equation to determine the length of any other diameter, of the form p*=- R2, although this diameter cannot meet the curve, yet if we measure on it from the centre lengths+R, these lines may be conveniently spoken of as diameters of the hyperbola. The locus of the extremities of these diameters which do not meet the curve is, by changing the sign of p2 in the equation of the curve, at once found to be This is the equation of a hyperbola having the axis of y for the axis meeting it in real points, and the axis of x for the axis meeting it in imaginary points. It is represented by the dotted curve on the figure, and is called the hyperbola conjugate to the given hyperbola. 167. We proved (Art. 165) that the diameters answering to b tan@=± meet the curve at infinity; they are, therefore, the a same as the lines called, in Art. 154, the asymptotes of the curve. They are the lines CK, CL on the figure, and evidently separate those diameters which meet the curve in real points from those which meet it in imaginary points. It is evident also that two conjugate hyperbolæ have the same asymptotes. b a The expression tan◊ = ± enables us, being given the axes in magnitude and position, to find the asymptotes, for if we form a rectangle by drawing parallels to the axes through B and A, then the asymptote CK must be the diagonal of this rectangle, But, since the asymptotes make equal angles with the axis of the angle which they make with each other must be = 20. Hence, being given the eccentricity of a hyperbola, we are given the angle between the asymptotes, which is double the angle whose secant is the eccentricity. Ex. To find the eccentricity of a conic given by the general equation. We can (Art. 74) write down the tangent of the angle between the lines denoted by ax2 + 2hxy + by2 = 0, and thence form the expression for the secant of its half; or we may proceed by the help of Art. 157, Ex. 3. 168. We now proceed to investigate some of the properties of the ellipse and hyperbola. We shall find it convenient to consider both curves together, for, since their equations only differ in the sign of b', they have many properties in common which can be proved at the same time, by considering the sign of b as indeterminate. We shall, in the following Articles, use the signs which apply to the ellipse. The reader may then obtain the corresponding formulæ for the hyperbola by changing the sign of b We shall first apply to the particular form a2 + y2 = b2 1, some of the results already obtained for the general equation. Thus (Art. 86) the equation of the tangent at any point x'y' being got by writing x'x and y'y for x2 and y* is The proof given in general may be repeated for this particular The equation of the chord joining any two points on case. which, when x', y=x", y", becomes the equation of the tangent already written. The argument here used applies whether the axes be rectangular or oblique. Now if the axes be a pair of conjugate diameters, the coefficient of xy vanishes (Art. 143); the coefficients. of x and y vanish, since the origin is the centre; and if a and b be the lengths of the intercepts on the axes, it is proved exactly, as in Art. 160, that the equation of the curve may be written And it follows from this article that in the same case the equation of the tangent is 169. The equation of the polar, or line joining the points. of contact of tangents from any point x'y', is similar in form to the equation of the tangent (Arts. 88, 89), and is therefore the axes of coordinates in the latter case being any pair of conjugate diameters, in the former case the axes of the curve. In particular, the polar of any point on the axis of x is = xx = 1. Hence the polar of any point P is found by drawing a diameter through the point, taking CP.CP' to the square of the semidiameter, and then drawing through P' a parallel to the conjugate diameter. This includes, as a particular case, the theorem proved already (Art. 145), viz., The tangent at the extremity of any diameter is parallel to the conjugate diameter. Ex. 1. To find the condition that Ax + y = 1 may touch xx' yy' y2 Comparing + = 1, λx+μy = 1, we find = λa, 1⁄2/ = μb, and a2X2 + b2μ2 = 1. a Ex. 2. To find the equation of the pair of tangents from x'y' to the curve (see Art. 92). Ans. (+1)(+1)=(+/-1)* Ex. 3. To find the angle between the pair of tangents from x'y' to the curve. When an equation of the second degree represents two right lines, the three highest terms being put = 0, denote two lines through the origin parallel to the two former; hence, the angle included by the first pair of right lines depends solely on the three highest terms of the general equation. Arranging, then, the equation found in the last Example, we find, by Art. 74, Ex. 4. Find the locus of a point, the tangents through which intersect at right angles. Equating to 0 the denominator in the value of tan, we find x2 + y2 = a2 + b2, the equation of a circle concentric with the ellipse. The locus of the intersection of tangents which cut at a given angle is, in general, a curve of the fourth degree. 170. To find the equation, referred to the axes, of the diameter conjugate to that passing through any point x'y' on the curve. The line required passes through the origin, and (Art. 169) is parallel to the tangent at x'y'; its equation is therefore Let 0, 0' be the angles made with the axis of x by the original diameter and its conjugate; then plainly tane; and from the equation of the conjugate we have (Art. 21) tan 0= Hence tan tan ✪ =- as might also be inferred from Art. 143. The corresponding relation for the hyperbola (see Art. 168) is 171. Since in the ellipse tane tan e' is negative, if one of the angles, be acute (and, therefore, its tangent positive), the other must be obtuse (and, therefore, its tangent negative). Hence, conjugate diameters in the ellipse lie on different sides of the axis minor (which answers to 90°). = In the hyperbola, on the contrary, tane tane' is positive; therefore and ' must be either both acute or both obtuse. Hence, in the hyperbola, conjugate diameters lie on the same side of the conjugate axis. b a' a In the hyperbola, if tan be less, tan e' must be greater than but (Art. 167) the diameter answering to the angle whose b tangent is is the asymptote, which (by the same Article) separates those diameters which meet the curve from those which do not intersect it. Hence, if one of two conjugate diameters meet a hyperbola in real points, the other will not. Hence also it may be seen that each asymptote is its own conjugate. 172. To find the coordinates x"y" of the extremity of the diameter conjugate to that passing through x'y'. These coordinates are obviously found by solving for x and y between the equation of the conjugate diameter and that of the curve, viz. |