document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=309192>Question 309192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Identify the conic. Choose the conic and its center. <BR>\n" );
document.write( "16x^2 + 25y^2 - 96x - 200y = -144 \r<BR>\n" );
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document.write( "I dont understand how to do this problem, ive tried and tried many times but i cant seem to hack it. Could you please help me step by step to solve this problem?  Thank you so very much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #221066 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=221066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You have both an x-squared term and a y-squared term, therefore it is NOT a parabola.\r<BR>\n" );
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document.write( "The signs on the x-squared and the y-squared terms are the same, therefore it is NOT a hyperbola.\r<BR>\n" );
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document.write( "The coefficients on the x-squared and y-squared terms are unequal, therefore it is NOT a circle.\r<BR>\n" );
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document.write( "Therefore it is an ellipse.  Since the coefficient on the x-squared term is smaller than the coefficient on the y-squared term, the ellipse has a horizontal major axis.\r<BR>\n" );
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document.write( "In order to determine the center, you need to put the equation into Standard Form.  Standard Form for an ellipse with a horizontal major axis and a center at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(h, k\right)\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(x\,-\,h)^2}{a^2}\ +\ \frac{(y\,-\,k)^2}{b^2}\ =\ 1\">\r<BR>\n" );
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document.write( "The process is called completing the square.\r<BR>\n" );
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document.write( "Arrange the equation so that the x-terms are together and the y-terms are together.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16x^2\ -\ 96x\ +\ 25y^2\ -\ 200y\ =\ -144\">\r<BR>\n" );
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document.write( "Factor out the lead coefficient from each part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16(x^2\ -\ 6x)\ +\ 25(y^2\ -\ 8y)\ =\ -144\">\r<BR>\n" );
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document.write( "Take the coefficient of the 1st degree x-term, divide by 2, square the result, add that result inside of the parentheses with the other x-terms, then add that result times the lead coefficient from the x-terms to the RHS.  -6 divided by 2 is -3.  -3 squared is 9.  Add 9 inside of the parentheses with the x-terms, and add 16 times 9 = 144 to the RHS of the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16(x^2\ -\ 6x\ +\ 9)\ +\ 25(y^2\ -\ 8y)\ =\ -144\ +\ 144\">\r<BR>\n" );
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document.write( "Repeat the process with the 1st degree y-term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16(x^2\ -\ 6x\ +\ 9)\ +\ 25(y^2\ -\ 8y\ +\ 16)\ =\ -144\ +\ 144\ + 400\">\r<BR>\n" );
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document.write( "Factor both trinomials:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16(x\ -\ 3)^2\ +\ 25(y\ -\ 4)^2\ =\ 400\">\r<BR>\n" );
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document.write( "Divide through by the constant in the RHS\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(x\ -\ 3)^2}{25}\ +\ \frac{(y\ -\ 4)^2}{16}\ =\ 1\">\r<BR>\n" );
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document.write( "Hence, the center is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(3, 4\right)\">  While we are at it, recognize that the the semi-major axis measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{25}\ =\ 5\"> and the semi-minor axis measures <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sqrt{16}\ =\ 4\">.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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