18. To find the magnitudes of the axes of the conic. Let a, B, y be the co-ordinates of the centre; and, for shortness' sake, put a-a=x, B-B=y, y-7=z. Then if r be the semi-diameter drawn from the centre to a, ß, y, we have (see Art. 3, Chap. 1.) (a cos A. x2 + b cos B. y2+c cos C.23)..............(1). Again, from the equation of the conic, 0 = p (a, B, y) = & (α+x, B+y, y+z) = & (a, B, y) + 2x (ua+w'ß+v'y) +2y (w'a+vß + u'y) + 2z (v'à + u'ß + wy) + $ (x, y, z). Now, by Art. 11 of the present chapter, ua+w'ß+v'\ _ w'a+vß+u'ŋ _ v'a+u'ß+wy Also, ax+by+cza (a− a) + b (B−B) + c (y-7)=0....(2) ; or, ux2+ vy2+ wz2 + 2u'yz + 2v'zx+2w'xy Now the semi-axes are the greatest and least values of the semi-diameter. We have then to make 442 abc p2 = a cos A. x2+b cos B. y+c cos C. z2........(4) a maximum or minimum, x, y, z being connected by the relations (2) and (3). Multiply (2) by the indeterminate multiplier 2, (4) by μ, adding them to (3), differentiating, and equating to zero the coefficients of each differential, we get (5). ux + w'y + v'z +λa cos A. x + μa = 0) có xtoy tra trỏ cos B .y + b = 0 v'x + u'y + wz +λc cos C.z + μc Multiplying these equations in order by x, y, z, and add ing, we get u, w', v' w', v, u' v', u', w u, w', v', = 0) w', v, u', b v', u', w, c a, b, C, 0 Substituting this value of A in equations (5), and eliminating x, y, z from the equations combined with (2), we obtain the following quadratic for the determination of 1 2.2 : 19. To find the area of the conic. In the above equation, the coefficient ofis - abcs2 (a cos B cos C+ b cos C cos A + cos A cos B), Hence the product of the two values of 2 is 442s2 u, w', v', a 'w', v, u', b v', u', w, c a, b, c, From the above investigation may be obtained the criterion which determines whether the conic be an ellipse or hyperbola. For, in the hyperbola, the two values of 2 have opposite signs, hence the curve will be an ellipse or hyperbola according as is negative or positive; or according as Ua2+ Vb2+ We+2 U'bc + 2 V'ca + 2 W'ab is positive or negative. 1. EXAMPLES. Each angular point of a triangle is joined with each of two given points; prove that the six points of intersection of the joining lines with the opposite sides of the triangle lie in a conic. 2. A conic is described, touching three given straight lines and passing through a given point; prove that the locus of its centre is a conic. Express, in geometrical language, the position of the given point relatively to the straight lines, in order that the locus of the centre may be a circle. Also find the locus of the given point, in order that the locus of the centre may be a rectangular hyperbola. 3. In example 1, prove that, if the conic described about the triangle, and passing through the two given points, touch the line (l, m, n), the conic passing through the six points of intersection will touch the line G 1 m , 4. If A, B, C, A', B', C' be six points, such that the straight lines B'C', C'A', A'B' are the several polars of the points A, B, C, with respect to a given conic, prove that The three straight lines AA', BB, CC', intersect in a point; and that The points of intersection of BC with B'C', CA with C'A', AB with A'B', lie in a straight line. 5. If two triangles circumscribe a conic, their angular points lie in another conic. 6. The equation of a conic circumscribing the triangle of reference, and having its semi-diameters parallel to the sides equal to r1, 1⁄2, r2 respectively, is 19 2) 3 7. A conic always touches the sides of a given triangle; prove that, if the sum of the squares on its axes be given, the locus of its centre is a circle, the centre of which is the point of intersection of the perpendiculars let fall from the angular points of the triangle on the opposite sides. 8. If be the angle between the asymptotes of the conic, represented by the general equation of the second degree, prove that |