document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3448>Question 3448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1  Eliminate, x,y between the following equations<BR>\n" );
document.write( "    x^2-y^2=px-qy, 4xy=qx+py , x^2+y^2=1\r<BR>\n" );
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document.write( "2. Eliminate, x,y,z between equations<BR>\n" );
document.write( "    y/z - z/y = a , z/x -x/z = b , x/y - y/x =c </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1537 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=1537\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Hi, somehow this is not an easy problem about elimination.\r<BR>\n" );
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document.write( " This first one should use formulas in Trig., for otherwise, it is<BR>\n" );
document.write( " almost impossible to get the cancellation of x,y & z.\r<BR>\n" );
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document.write( "1  Eliminate, x,y between the following equations<BR>\n" );
document.write( "    x^2-y^2=px-qy...(1), 4xy=qx+py...(2) , x^2+y^2=1...(3).<BR>\n" );
document.write( "   Sol: Since (3) is the unit circle x^2 + y^2=1,<BR>\n" );
document.write( "        x = cos t, y = sin t for some t.<BR>\n" );
document.write( "        Goto (1), we have cos^2 t- sin^2 t = (By double angle formula)<BR>\n" );
document.write( "        cos 2t = p cos t - q sin t = sqrt(p^2+q^2) [p cos t/sqrt(p^2+q^2) - qsin t/sqrt(p^2+q^2)]<BR>\n" );
document.write( "        = sqrt(p^2+q^2) cos (w + t)....(4)  where cos w = p/sqrt(p^2+q^2)<BR>\n" );
document.write( "        Goto (2),  4 cos t sin t = q cos t + p sin t <BR>\n" );
document.write( "        = sqrt(p^2+q^2) [q cos t/sqrt(p^2+q^2) + p sin t/sqrt(p^2+q^2)]<BR>\n" );
document.write( "      So, 2 sin 2t = sqrt(p^2+q^2) sin (w + t) ...(5)  [Note cos w = p/sqrt(p^2+q^2)]<BR>\n" );
document.write( "      <BR>\n" );
document.write( "      (4)^2 + (5)^2 : cos^2 2t + 4 sin^2 2t = (p^2+q^2) [cos^2 (w + t) + sin^2 (w + t)]<BR>\n" );
document.write( "      Simplify both sides: 1 + 3 sin^2 2t = p^2 + q^2 [we can get the same relation by using (1)^2+(2)^2 ]<BR>\n" );
document.write( "      or sin^2 2t = (p^2 + q^2 - 1)/3 or 4 sin^2 t cos^2 t = (p^2 + q^2 - 1)/3\r<BR>\n" );
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document.write( "      Since sin^2 2t = 4 (sin^2 t/ cos^2 t) cos^4 t = 4 tan^2 t /sec ^4 t = 4 tan^2 t /(1+tan^2 t )^2,<BR>\n" );
document.write( "      we have  tan^2 t /(1+tan^2 t )^2 = (p^2 + q^2 - 1)/12 or<BR>\n" );
document.write( "       [(1+tan^2 t )^2/tan^2 t] = 12/(p^2 + q^2 - 1) or<BR>\n" );
document.write( "      So, (1+tan^2 t )/tan t = sqrt[12/(p^2 + q^2 - 1)] or<BR>\n" );
document.write( "      tan t + 1/tan t = sqrt[12/(p^2 + q^2 - 1)] ...(6)<BR>\n" );
document.write( "      Let r = sqrt[12/(p^2 + q^2 - 1)] = 2sqrt[3/(p^2 + q^2 - 1)]<BR>\n" );
document.write( "    Convert (6) to the quuadratic equation  u^2 -ru + 1 = 0 , where u = tan t,<BR>\n" );
document.write( "   by quadratic formula u = tan t = [r + sqrt(r^2-4)]/2 [or (r - sqrt(r^2-4))/2]<BR>\n" );
document.write( "   Hence, tan t = [r+sqrt(r^2-4)]/2 = u ...(7)<BR>\n" );
document.write( "   Consider  (5)/(4): 2 tan 2t = tan (w+t), so<BR>\n" );
document.write( "         4 tan t/(1 - tan^2 t) = (tan w + tan t) / (1 - tan w tan t)<BR>\n" );
document.write( "      Cancelling the denominator and simplify:<BR>\n" );
document.write( "        tan^3 t - 3 tan w tan ^2 t + 3 tan t - tan w = 0 ...(8)  <BR>\n" );
document.write( "     Replace tan t by (7) in (8) , we obtain <BR>\n" );
document.write( "  [Note tan w = q/p, and so tan t = y/x , x, y disappear.]<BR>\n" );
document.write( " <BR>\n" );
document.write( "  [r+sqrt(r^2-4)]^3/8 - 3 q/p^2/4 + 3[r+sqrt(r^2-4)]/2 - q/p =0.\r<BR>\n" );
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document.write( " {Recall : r =  2sqrt[3/(p^2 + q^2 - 1)] ,r^2 = 12/(p^2 + q^2 - 1)}<BR>\n" );
document.write( " {You can try to simplify it. But, it seems that there is no simple algebraic<BR>\n" );
document.write( " form of for it. In other words, it is in complicated and ugly form.]]\r<BR>\n" );
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document.write( " 2. Eliminate, x,y,z between equations<BR>\n" );
document.write( "    y/z - z/y = a , z/x -x/z = b , x/y - y/x =c.\r<BR>\n" );
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document.write( "  Let y/z = u, z/x = w,  x/y = wu.<BR>\n" );
document.write( "  So, u -1/u = a,<BR>\n" );
document.write( "   or  u^2 - au -1 =0<BR>\n" );
document.write( "  By quadratic formula  u = [a+ sqrt(a^2+4)]/2 = y/z...(1) <BR>\n" );
document.write( "  [if u=y/z >0, otherwise  u= (a- sqrt(a^2+4))/2 ]<BR>\n" );
document.write( " <BR>\n" );
document.write( "   Similary, w = [b+ sqrt(b^2+4)]/2 = z/x...(2)  [if w=z/x >0]<BR>\n" );
document.write( "    and x/y = [c + sqrt(c^2+4)]/2 ...(3) [if x/y >0]<BR>\n" );
document.write( "  <BR>\n" );
document.write( " Since y/z * z/x * x/y = 1, we obtain<BR>\n" );
document.write( "   [a + sqrt(a^2+4)] [b + sqrt(b^2+4)][c+ sqrt(c^2+4)]/8 = 1,<BR>\n" );
document.write( "  [a + sqrt(a^2+4)] [b + sqrt(b^2+4)][c+ sqrt(c^2+4)] = 8.\r<BR>\n" );
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document.write( " More precisely,<BR>\n" );
document.write( " Case(i) y/z, z/x > 0 <BR>\n" );
document.write( "  [a + sqrt(a^2+4)] [b + sqrt(b^2+4)][c+ sqrt(c^2+4)] = 8.<BR>\n" );
document.write( " Case(ii) y/z > 0, z/x < 0 <BR>\n" );
document.write( "  [a + sqrt(a^2+4)] [b - sqrt(b^2+4)][c- sqrt(c^2+4)] = 8. <BR>\n" );
document.write( " Case(ii) y/z < 0, z/x > 0<BR>\n" );
document.write( "  [a - sqrt(a^2+4)] [b + sqrt(b^2+4)][c- sqrt(c^2+4)] = 8. <BR>\n" );
document.write( " Case(iii) y/z < 0, z/x < 0<BR>\n" );
document.write( "  [a - sqrt(a^2+4)] [b - sqrt(b^2+4)][c+ sqrt(c^2+4)] = 8. \r<BR>\n" );
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document.write( " Try to read carefully about every step.<BR>\n" );
document.write( " Good luck !\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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