A similar mode of proof is applicable to the hyperbola. COR. If the axes be rectangular, AT.Vt= BC2, (fig. 57.) Hence, in the circle, AT. Bt=r2. 117. PROP. 8. If CP, CD, be a given system of semiconjugate diameters, and CP, CD' any other system, then if a tangent at P meet CP, CD' in the points T, t, PT. Pt=CD2. For CP, CD being assumed as the axes, the equations to CP', CD' will be It may, in like manner be shewn, that in the hyperbola, PT. Pt+b2. The reason why the sign of the product is negative in the ellipse, and positive in the hyperbola, is, that PT, Pt lie on different sides of CP in the former, and on the same side of it, in the latter. COR. If the conjugate diameters which the tangent meets be the axes, then PT. Pt=CD3. (fig. 58.) In the circle, PT. Pt=CP', (fig. 59.) 118. PROP. 9. To draw a tangent to the ellipse and hyperbola from a given point (a, B) without the curve. Let the curve be referred to its principal diameters; then if x', y' be the co-ordinates of the point of contact, a2yy' + b2xx' = a2b2 will be the equation to the tangent; x', and y' being supposed unknown. Now a, ß being the co-ordinates of a point in the tangent, these quantities, when substituted for x and y, will satisfy the above equation; .. a2y'. ß+b2x'. a= a2b2 . . . . . .(1). But since (x, y) is a point in the curve, we have a2y'2+b2x2=a2b2. . . .. . (2). Hence, the quantities sought, x' and y', will be obtained, by elimination, from equations (1) and (2): and since the latter is a quadratic, there will be two points of contact; in other words, two tangents to the ellipse or hyperbola, may be drawn from a given point without the curve. It will be easier, however, instead of actually combining these equations, to construct the loci to which they belong, since their intersection will evidently determine the points of contact. Now the locus of equation (2) is the curve itself, and since equation (1) represents a straight line, it can be no other than the line which joins the points of contact. Whence the two tangents may thus be drawn. Let Q (fig. 60.) be the given point (a, ß), then in the equation a2y'.ẞ+b2x'. a=a2b3., draw T't meeting the curve in the points P, p, then, the lines QP, Qp will be the tangents required. CHAP. IV. ON THE PROPERTIES OF CONJUGATE DIAMETERS IN THE ELLIPSE AND HYPERBOLA. 119. PROP. 1. ANY two conjugate diameters of known inclination being given, to find the magnitude and direction of the principal diameters. Let 2a, 2b' be the conjugate diameters, y the angle which they form, and suppose the curve referred to these di ameters as axes. It was shewn in Art. 115, that the tangent to an ellipse, at the extremity of either of the principal diameters coincides with the tangent at the same point in the circle described on that diameter. This property suggests a simple method of resolving the proposed question. Let x', y' be the co-ordinates of the extremity of either principal diameter, and 2r its length. Then the equation to the tangent at (x', y') in the ellipse is ayy' + b2xx'=a2b3...... (1), and that to the tangent at the same point in the circle described upon 2r is (y + x' cos y) y + (x + y cos y) x = r2...... (2), and since the tangents represented by these equations coincide, we have, equating the corresponding terms, hence, multiplying out and arranging the result, which equation being resolved, These two values of r2 being always real and positive, there are in the ellipse four values of r, two of which are equal, with opposite signs, to the other two. real principal diameters. b2 Hence, the ellipse has two Also, = diameter is known. In the hyperbola, a and b have different signs; whence one of the values of r is negative, and two values therefore of r imaginary. The hyperbola therefore has only one real principal diameter. 120. COR. 1. Let a, b, be the roots of equation (3), then, by the theory of equations, we have a2+b2 = a'2 +b22...........(1), ab a'b' sin y..........(2). From the former of these equations we infer, that “in the ellipse, the sum of the squares of any two conjugate diameters |