document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=581877>Question 581877</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the following equation in standard form, then identify the type of conic and graph:\r<BR>\n" );
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document.write( "64x^2 + 49y^2 +256x - 196y - 2684 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #371997 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=371997\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "You can determine the type of conic by simply examining the given equation.\r<BR>\n" );
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document.write( "Both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> terms exist, so the conic is NOT a parabola.\r<BR>\n" );
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document.write( "The coefficients on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> terms are unequal, so the conic is NOT a circle.\r<BR>\n" );
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document.write( "The coefficients on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> terms have the same sign, so the conic is NOT a hyperbola.\r<BR>\n" );
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document.write( "The conic is therefore an ellipse.\r<BR>\n" );
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document.write( "Complete the square on each variable.\r<BR>\n" );
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document.write( "Step 1:  Constant term to the RHS.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64x^2\ +\ 49y^2\ +\ 256x\ -\ 196y\ =\ 2684\">\r<BR>\n" );
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document.write( "Step 2:  Gather like variables:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(64x^2\ +\ 256x\right)\ +\ \left(49y^2\ -\ 196y\right)\ =\ 2684\">\r<BR>\n" );
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document.write( "Step 3:  Factor lead coefficient from each of the binomials.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64\left(x^2\ +\ 4x\right)\ +\ 49\left(y^2\ -\ 4y\right)\ =\ 2684\">\r<BR>\n" );
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document.write( "Step 4:  Complete the square on each binomial.  Divide the 1st order term coefficient by 2, square the result, add that result inside of the parentheses.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64\left(x^2\ +\ 4x\ +\ 4\right)\ +\ 49\left(y^2\ -\ 4y\ +\ 4\right)\">\r<BR>\n" );
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document.write( "Step 5:  Take the number added inside of the parentheses times the coefficient outside of the parentheses and add it to the LHS to compensate.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64\left(x^2\ +\ 4x\ +\ 4\right)\ +\ 49\left(y^2\ -\ 4y\ +\ 4\right)\ =\ 2684\ +\ 256\ +\ 196\ =\ 3136\">\r<BR>\n" );
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document.write( "Step 6:  Factor the perfect square trinomials in the RHS:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 64\left(x\ +\ 2\right)^2\ +\ 49\left(y\ -\ 2\right)^2\ =\ 3136\">\r<BR>\n" );
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document.write( "Step 7:  Divide by the coefficient 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\ +\ 2)^2}{49}\ +\ \frac{(y\ -\ 2)^2}{64}\ =\ 1\">\r<BR>\n" );
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document.write( "The standard form of the ellipse 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( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\ >\ b\">, the semi-major axis is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> and is parallel to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> axis, the semi-minor axis is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\">, the center is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h, k)\">, the vertices are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (-a\,+\,h,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a\,+\,h,k)\">, the minor axis endpoints are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,-b\,+\,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,b\,+\,k)\">, and the foci are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (-c\,+\,h,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (c\,+\,h,k)\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c\ =\ \sqrt{a^2\ -\ b^2}\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(x\ -\ h)^2}{b^2}\ +\ \frac{(y\ -\ k)^2}{a^2}\ =\ 1\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\ >\ b\">, the semi-major axis is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> and is parallel to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\"> axis, the semi-minor axis is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b\">, the center is at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h, k)\">, the vertices are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,-b\,+\,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,b\,+\,k)\">, the minor axis endpoints are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (-a\,+\,h,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a\,+\,h,k)\">, and the foci are at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,-c\,+\,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (h,c\,+\,k)\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c\ =\ \sqrt{a^2\ -\ b^2}\"><BR>\n" );
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document.write( "Re-write your equation so that the key values can be seen by inspection.  The graph should be obvious from that point:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{(x\ -\ (-2))^2}{7^2}\ +\ \frac{(y\ -\ 2)^2}{8^2}\ =\ 1\"><BR>\n" );
document.write( "  \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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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