College algebra . does not intersect the circle. 1?; x2 + 2/2 = r\ 18. 0^ + 2/2 = 25, x + y = W. 3x + 4 2/ = c. 19. iE2 + 2/^ = 25, 20. a;2 + 2^ = 25, ax + 6y = 1. T X — by = 3- 21. For what values of 6 in terms of r and m does the system of equations 2/ = mx + b,a;2 + 2/2 = r2have equal solutions ? 22. Determine the relation between a, b, and k such that the system y = mx + k, ^^=1a2 62has equal solutions. 80 SIMULTANEOUS QUADRATICS [Chap. VII. Case III. When all the terms which contain the unknowns areof the second degree. > EXERCISES Solve the following pairs of a^ + 3 ai
College algebra . does not intersect the circle. 1?; x2 + 2/2 = r\ 18. 0^ + 2/2 = 25, x + y = W. 3x + 4 2/ = c. 19. iE2 + 2/^ = 25, 20. a;2 + 2^ = 25, ax + 6y = 1. T X — by = 3- 21. For what values of 6 in terms of r and m does the system of equations 2/ = mx + b,a;2 + 2/2 = r2have equal solutions ? 22. Determine the relation between a, b, and k such that the system y = mx + k, ^^=1a2 62has equal solutions. 80 SIMULTANEOUS QUADRATICS [Chap. VII. Case III. When all the terms which contain the unknowns areof the second degree. > EXERCISES Solve the following pairs of a^ + 3 aiy = 28,a;2 + 2/2 = t Let y = mx and substitute in both, the first equation, we have a;2 + 3 mx^ = 28,28 whence x^ =,: Prom the second equation, we have whence x^ Equating these values of x?, we obtain 28 l + 3m :20, 20 1 + »l220 1+ 3 TO 1 + m2Clearing of fractions and reducing, we obtainf 7m2 —15m + 2 = 0, or m = 2, or \. 28 Substituting these values of m inwe find, for to = 2, X2:. FiQ. 14. l + Snia;2 = 4, a; = ±2,y = ma; = ± ;2 = s/, a; = ± -jVlO = ± ,y = mx = ±l VlO = ± +. The solutions are therefore(2, 4), (-2, -4), (JVlO,Jv^), (-|VlO, -JVlO). The loci of the two equationsof this exercise are shown in>-X Fig. 14. The geometrical inter-pretation of the substitutiony — mx is also shown in thefigure. 2. a;S + 3zs/ = 28,4 J/ + xj/ = 8. 3. xs + xj/ + 3/2 = 6,x2 + 3/S = 12. Art. 53] SOLUTION OF SIMULTANECtoS QUADRATICS 81 4. 2y^-^xy + 3x^=n, S. a;2 + 2/2 = 6&, 2/2 _ aj2 = 16. !ej/ = 28. ^ 6. x^ + xv + 2y^ = li, 7. x^-42/2 = 9, 2x^ + 2xy +y^ = 13. xy + 2y^ = 3. e_x + y^x^W 9. x2_X2/ + 2/2 = 21. r7,/V 2/2-2x2/ + 15 = 0. x^ + y^ = 45. 10. 4 a2 - 2 o5 = 62 - 16, 11. x^ + xy = 4, 5 o2 = 7 a6 — 36. j/2 + a;j/ = 1. Find to two significant figures the solutions of the following : 12. a2 + 2/2 = , 13. a;2 + ;s^= 104, a^ = yi + lAxy = 21. Case IV. When the equations are symmetrical.*The typical form of a symmetrical
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