document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=384600>Question 384600</A>: <I><SPAN STYLE=\"word-break: break-all;\"> identify the conic section. if it is a parabola, give the vertex. if it is a circle, gibe the center and radius. if it is an ellipse or a hyperbola, give the center and foci.<BR>\n" );
document.write( "4x2 + 7y2 - 40x + 42y +135 =0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #272310 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=272310\">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( "There is an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> term AND a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> term -- not a parabola.\r<BR>\n" );
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document.write( "The signs 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 the same -- not a hyperbola.\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 not equal -- not a circle.\r<BR>\n" );
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document.write( "Therefore, it is an ellipse.\r<BR>\n" );
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document.write( "Complete the squares on each of the variables.\r<BR>\n" );
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document.write( "First move the constant term to the RHS and group like variables:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ -\ 40x\ +\ 7y^2\ +\ 42y\ =\ -135\">\r<BR>\n" );
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document.write( "Complete the square on <i>x</i>:\r<BR>\n" );
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document.write( "Factor the coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> term out of 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 x\"> terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x^2\ -\ 10x\right)\ +\ 7y^2\ +\ 42y\ =\ -135\">\r<BR>\n" );
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document.write( "Divide the coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> by 2, square the result, and add that result to the LHS <i>inside</i> the parentheses, then add the lead coefficient times the result of squaring half of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> coefficient to the RHS.  -10 divided by 2 is -5.  -5 squared is 25.  Add 25 inside the parentheses on the left and 100 to the right.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x^2\ -\ 10x\ +\ 25\right)\ +\ 7y^2\ +\ 42y\ =\ -135\ +\ 100\">\r<BR>\n" );
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document.write( "Repeat the same process for the <i>y</i> terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x^2\ -\ 10x\ +\ 25\right)\ +\ 7\left(y^2\ +\ 6y\ +9\right)\ =\ -135\ +\ 100\ +\ 63\">\r<BR>\n" );
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document.write( "Factor the two perfect square trinomials in the LHS and do the arithmetic in the RHS:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x\ -\ 5\right)^2\ +\ 7\left(y\ +\ 3\right)\ =\ 28\">\r<BR>\n" );
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document.write( "Divide both sides by the RHS constant.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\left(x\ -\ 5\right)^2}{7}\ +\ \frac{\left(y\ +\ 3\right)}{4}\ =\ 1\">\r<BR>\n" );
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document.write( "Since there was no <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large xy\"> term in the original equation, the axes of the ellipse are parallel to the coordinate axes.  Since the denominator in the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term is larger than the denominator in the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> term, the major axis is parallel to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> axis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\left(x\ -\ h\right)^2}{a^2}\ +\ \frac{\left(y\ -\ k\right)}{b^2}\ =\ 1\">\r<BR>\n" );
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document.write( "Is an equation of an ellipse with center at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (h, k)\">\r<BR>\n" );
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document.write( "Semi-major axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">, semi-minor axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">\r<BR>\n" );
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document.write( "Vertices:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (h\,\pm\,a, k)\">\r<BR>\n" );
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document.write( "Foci:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (h\,\pm\,c, k)\">\r<BR>\n" );
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document.write( "where\r<BR>\n" );
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document.write( "<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( "And while we are at it, the endpoints of the minor axis segement are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (h,k\,\pm\,b)\">\r<BR>\n" );
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document.write( "For your problem, you will find it convenient to rewrite the standard form equation thusly:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\left(x\ -\ 5\right)^2}{(\sqrt{7})^2}\ +\ \frac{\left(y\ -\ (-3)\right)}{2^2}\ =\ 1\">\r<BR>\n" );
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document.write( "You can see that the center is at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (5, -3)\">\r<BR>\n" );
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document.write( "Then, to fix the foci, calculate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ c\ =\ \sqrt{7\ -\ 4}\ =\ \sqrt{3}\">\r<BR>\n" );
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document.write( "So the foci are at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (5\,\pm\,\sqrt{3}, -3)\">\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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