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 bas 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 b2. We shall first apply to the particular form a2 y2 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 x and y" is xx a2 y'y +33 = 1. 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 х 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 = a 72 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 λx + μy = 1 may touch x2 y2 = 1. ! = λa, % = μb, and a2X2 + b2μ2 = 1. Ex. 2. To find the equation of the pair of tangents from x'y' to the curve (see Art. 92). Ex. 3. To find the angle y'2 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, y'2 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 tano, 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 tan@= 2; and from = the equation of the conjugate we have (Art. 21) tan0= Hence tan tan 0: a2 as might also be inferred from Art. 143. The corresponding relation for the hyperbola (see Art. 168) is 171. Since in the ellipse tane tane is negative, if one of the angles 0, ' 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, tan 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' In the hyperbola, if tan be less, tan e' must be greater than but (Art. 167) the diameter answering to the angle whose tangent is is the asymptote, which (by the same Article) b α 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'. y These coordinates are obviously found by solving for x and between the equation of the conjugate diameter and that of the curve, viz. Substituting in the second the values of x and y found from the first equation, and remembering that x', y' satisfy the equation of the curve, we find without difficulty 173. To express the lengths of a diameter (a'), and its conjugate (b'), in terms of the abscissa of the extremity of the diameter. (1) We have But Hence a" = x2 + y2. or, The sum of the squares of any pair of conjugate diameters of an ellipse is constant (see Ex. 3, Art. 159). 174. In the hyperbola we must change the signs of b2 and b'2, and we get a22 — b” — a2 — b3, = or, The difference of the squares of any pair of conjugate diameters of a hyperbola is constant. If in the hyperbola we have ab, its equation becomes x2 - y2 = a3, and it is called an equilateral hyperbola. The theorem just proved shows that every diameter of an equilateral hyperbola is equal to its conjugate. The asymptotes of the equilateral hyperbola being given by the equation are at right angles to each other. Hence this hyperbola is often called a rectangular hyperbola. The condition that the general equation of the second degree should represent an equilateral hyperbola is a =-b; for (Art. 74) this is the condition that the asymptotes (ax + 2hxy + by*) should be at right angles to each other; but if the hyperbola be re tangular it must be equilateral, since (Art. 167) the tangent b of half the angle between the asymptotes =; therefore, if this angle = 45°, we have b =α. 175. To find the length of the perpendicular from the centre on the tangent. The length of the perpendicular from the origin on the line 176. To find the angle between any pair of conjugate dia meters. The angle between the diameters is equal to the angle be tween either, and the tangent parallel to T = The equation a'b' sino ab proves that the triangle formed by joining the extremities of conjugate diameters of an ellipse or hyperbola has a constant area (see Art. 159, Ex. 2). |