document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209670>Question 1209670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the constant k so that the equation<BR>\n" );
document.write( "4x^2 + 9y^2 - 8x + 54y + k = 2x^2 + 5y^2 - 12x + 34y<BR>\n" );
document.write( "represents an ellipse which has an area of 6 \pi.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849782 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849782\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the constant k:\r<BR>\n" );
document.write( "\n" );
document.write( "1. **Rearrange the equation:** Combine like terms and move everything to one side:\r<BR>\n" );
document.write( "\n" );
document.write( "   4x² + 9y² - 8x + 54y + k - 2x² - 5y² + 12x - 34y = 0<BR>\n" );
document.write( "   2x² + 4y² + 4x + 20y + k = 0\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Complete the square:**  Group the x and y terms and complete the square for each:\r<BR>\n" );
document.write( "\n" );
document.write( "   2(x² + 2x) + 4(y² + 5y) + k = 0<BR>\n" );
document.write( "   2(x² + 2x + 1) - 2 + 4(y² + 5y + 25/4) - 25 + k = 0<BR>\n" );
document.write( "   2(x + 1)² + 4(y + 5/2)² + k - 27 = 0<BR>\n" );
document.write( "   2(x + 1)² + 4(y + 5/2)² = 27 - k\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Standard form of an ellipse:** Divide by (27 - k) to get the equation in standard form:\r<BR>\n" );
document.write( "\n" );
document.write( "   [2(x + 1)²] / (27 - k) + [4(y + 5/2)²] / (27 - k) = 1<BR>\n" );
document.write( "   [(x + 1)²] / [(27 - k)/2] + [(y + 5/2)²] / [(27 - k)/4] = 1\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Area of an ellipse:** The area of an ellipse is given by A = πab, where a and b are the semi-major and semi-minor axes. In our case:\r<BR>\n" );
document.write( "\n" );
document.write( "   a² = (27 - k)/2  => a = √[(27 - k)/2]<BR>\n" );
document.write( "   b² = (27 - k)/4  => b = √[(27 - k)/4] = √[(27 - k)/2]/√2 = a/√2\r<BR>\n" );
document.write( "\n" );
document.write( "5. **Solve for k:** We're given that the area is 6π:\r<BR>\n" );
document.write( "\n" );
document.write( "   6π = πab<BR>\n" );
document.write( "   6 = ab<BR>\n" );
document.write( "   6 = √[(27 - k)/2] * √[(27 - k)/4]<BR>\n" );
document.write( "   6 = (27 - k)/2√2<BR>\n" );
document.write( "   12√2 = 27 - k<BR>\n" );
document.write( "   k = 27 - 12√2\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the constant k is 27 - 12√2.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );