The coefficients of λ2 and of λ3 are both zero. Hence there is a simple equation for λ, and therefore from (i) we have for the equation of the asymptotes $ (α, B, y) u w', v', a +(ax+bB+cy)2 | u = : 0. พ с 268. To find the condition that the conic may be a rectangular hyperbola. Change to Cartesian co-ordinates. Then the conic will be a rectangular hyperbola if the sum of the coefficients of x2 and y2 is zero. The condition becomes u+v+w-2u' cos A – 2v' cos B-2w' cos C = 0. 269. To find the equation of the circle circumscribing the triangle of reference. If from any point P, on the circle circumscribing a triangle ABC, the three perpendiculars PL, PM, PN be drawn to the sides of the triangle and meet the sides BC, CA, AB in the points L, M, N respectively; then it is known that these three points L, M, N are in a straight line. Let the triangle be taken for the triangle of reference and let a, ẞ, y be the co-ordinates of P. The areas of the triangles MPN, NPL, and LPM are By sin A, ya sin B, and aß sin C respectively. Since L, M, N are on a straight line, one of these triangles is equal to the sum of the other two. Hence, regard being had to sign, we have or By sin A+ ya sin B+ aß sin C = 0, aßy + byx+cxß = 0, which is the equation required. Ex. The perpendiculars from O on the sides of a triangle meet the sides in D, E, F. Shew that, if the area of the triangle DEF is constant, the locus of O is a circle concentric with the circumscribing circle. 270. Since the terms of the second degree are the same in the equations of all circles, if S=0 be the equation of any one circle, the equation of any other circle can be written in the form or, S+λα + β + γ = 0, in the homogeneous form, S+ (la+mẞ+ny) (aa+bB+cy) = 0. 271. From the form of the general equation of a circle in Art. 270 it is evident that the line at infinity cuts all circles in the same two (imaginary) points. The two points at infinity through which all circles pass are called the circular points at infinity. Since, in Cartesian co-ordinates, the lines x+y=0 are parallel to the asymptotes of any circle, the imaginary lines x2+y=0 go through the circular points at infinity. Hence, from Art. 193, the four points of intersection of the tangents drawn to a conic from the circular points at infinity are the four foci of the curve. 272. To find the conditions that the curve represented by the general equation of the second degree may be a circle. The equation of the circle circumscribing the triangle of reference is [Art. 269] aBy+bya + caẞ=0. Therefore [Art. 270] the equation of any other circle is of the form aBy+bya + caẞ + (la+mẞ+ny) (aa+bB+cy) = 0. If this is the same curve as that represented by ua2 + vß2 + wy2+2u' By + 2v'ya + 2w'aß = 0, we must have, for some value of λ, 2λu' = a + cm + bn, 2λv' = b+an+cl,and 2λw' =c+bl+am. Hence 2bcu'-c'v-b3w=2cav' - a3w - c'u for each of these quantities is equal to = 2abw' - b3u — a2v, abc λ 273. To find the condition that the conic represented by the general equation of the second degree may be an ellipse, parabola, or hyperbola. The equation of the lines from the angular point C to the points at infinity on the conic will be found by eliminating y from the equation of the curve and the equation ax+bB+cy=0. Hence the equation of the lines through C parallel to the asymptotes of the conic will be uca + vỏ tàu (ax +BB) – 2c (ax +BB) -2v'cx (ax+bB) +2w'c'aẞ = 0. The conic is an ellipse, parabola, or hyperbola, according as these lines are imaginary, coincident, or real; and the lines are imaginary, coincident, or real according as (wab-u'ac - v'bc+w'c2)2 — (uc2 + wa2 — 2v′ ac) (vc2 + wb3 — 2u'bc) is negative, zero, or positive; that is, according as Ua2+ Vb2+ Wc2 + 2 Ubc + 2 V'ca +2 W'ab is positive, zero, or negative. 274. The equation of a pair of tangents drawn to the conic from any point can be found by the method of Art. 188. The equation of the director circle of the conic can be found by the method of Art. 189. The equation giving the foci can be found by the method of Art. 193. The equations for the foci will be found to be аф do)2 do с do 2 4 (b3w + c°v — 2bcu') $ (a, ß, y) – (bad α αβ -bd da αφ a dy 2 da. The elimination of p (a, B, y) will give the equation of the axes of the conic. 275. To find the equation of a conic circumscribing the triangle of reference. are The general equation of a conic is uï2 + vß2 + wy2 + Qu'ẞy + 2v'ya + 2w'aß = 0. The co-ordinates of the angular points of the triangle If these points are on the curve, we must have u = 0, v = 0, and w = 0, as is at once seen by substitution. Hence the equation of a conic circumscribing the triangle of reference is u'By + v'yz+w'aß = 0. 276. The condition that a given straight line may touch the conic may be found as in Art. 263, or as follows. The equation of the lines joining A to the points common to the conic and the straight line (l, m, n), found by eliminating a between the equations of the conic and of the straight line, is or lu' By — (v'y+w'ß) (mß + ny) = 0, mw'ß2 + nv'y2 + (mv' + nw' — lu') By = 0. The lines are coincident if (l, m, n) is a tangent; the condition for this is which is equivalent to √lu' ± √mv' ± √nw' = 0. 277. To find the equation of a conic touching the sides of the triangle of reference. The general equation of a conic is ux2 + vß2 + wy2 + Qu' By + 2v′yx + 2w'aß = 0. Where the conic cuts a = 0, we have vß2 + wy2 + 2u' By = 0. Hence, if the conic cut a=0 in coincident points we have vw=u22, or u' = √vw. Similarly, if the conic touch the other sides of the triangle, we have v' = √wu, and w' = √uv. Putting λ2, μ2, v2 instead of u, v, w respectively, we have for the equation, λ2a2 + μ3ß2 + v3y2 = 2μvßy F 2vXyz ‡ 2λμaß = 0. In this equation either one or three of the ambiguous signs must be negative; for otherwise the left side of the equation would be a perfect square, in which case the conic would be two coincident straight lines. The equation can be written in the form 278. To find the condition that the line la+mẞ+ny = 0 may touch the conic √λa + √μß + √vy = 0. The condition of tangency can be found as in Art. 276, the result is 279. To find the equations of the circles which touch the sides of the triangle of reference. If D be the point where the inscribed circle touches BC, we know that |