M = 2a*{Da> + 5)/L, N = a''((i)^ + GE^)/L. The corresponding values of y are then given by 

 y=±[G(x^-a^)]'/7a. 



Equation (3) may be solved by the standard algebraic method^ or by any of a number of 

 numerical techniques.' 



n. Again, if equations (l) and (2) are written in the forms x^ - Qy^ = a^, x^ + Uxy + Vy^ 

 + Wx + Ry + T = 0, where Q = a7(c' - a^'), U = B/A, V = C/A, W = D/A, R = E/A, T = F/A 

 and these forms of the equations solved simultaneously with the line of slope m through the 

 common focus F(c,o) whose equation is y = m(x - c), one obtains the two equations: 



(Qm' - 1) x' - 2cQm'x + (a' + c'Qm') = 0, (4) 



(1 +Um + Vm=')x'+[W + (R-cU)m-2cVm']x + (c'Vm'-cRm + T) = 0, 



The resultant of the quadratic equations (4) is the condition that they have the same 

 solutions or correspondingly that the parameter m will be restricted to those values for the 

 points common to the hyperbolas (1) and (2)/ 



The resultant of the quadratics aoX^ + a,x + aj = 0, b^x^ + b,x + bj = is given by 



(aobj - b^aj)^ + (b,a2 - a^bj) (aob, - aibo) = 0. (5) 



From (4) ao = Qm' - 1, a, = -2cOm', a, = a' + c'Qm% b^ = 1 +Um+Vm% 

 bj = [W + (R - cU) m - 2c Vm^] , b, = c^Vm^ - cRm + T, and these values placed in (5) lead 

 to the quartic equation 



kim" + kjm' + k^m^ + k4m + k; =0, (6) 



where with G = c^" - a', fi = (a' + c') V + (c' - T),6'o = R + cU, 0=c' + cW +T, 

 77=R-cU, ^=a'U-cR, p = a'-T, p'=a' + T one finds: k^ = (GV +c/)Qy-a'ei, 

 kj = 2[^fi + 277ca'V + a'RQ . (W + 2c) + c'QU(cW + 2T)], k, = ^' - aV + 2p'n + W[4a'cV + 

 2cpQ - a'W], k, = 2(p'f - a'WTy), k^ = p'^ - a'W\ 



Again the solutions of (6) may be found by well known algebraic or numerical methods. 

 The values of m obtained are of course the slopes of the lines through F{c,o) and the points 

 of intersection of the hyperbolas (1) and (2). 



^College Algebra, H. B. Fine, Page 486. 



'Numerical Mathematical Analysis, J. B. Scarborough, Second Edition, 1950, Pages 62—72. 

 (The Johns Hopkins Press, Baltimore) 



"College Algebra, H. B. Fine, Page 512. 



103 



