document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=383818>Question 383818</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Directions: For each of the following conic sections identify whether it is a circle, parabola, ellipse or hyperbola. Find its center and/or vertices, its focus or foci, major or transverse axis, if appropriate, asymptotes or directrix if present. Graph each of them, labeling all important points.\r<BR>\n" );
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document.write( "The question I'm having trouble with is:<BR>\n" );
document.write( "4x^2+y^2-24x+4y+36=0\r<BR>\n" );
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document.write( "So far, I've gotten this, but I think I did something wrong:<BR>\n" );
document.write( "4x^2+y^2-24x+4y+36=0<BR>\n" );
document.write( "[-4(x+3)^2-9]+y^2+4y+36=0<BR>\n" );
document.write( "[-4(x+3)^2-9]+(y+2)^2-4+36=0<BR>\n" );
document.write( "[-4(x+3)^2]+(y+2)^2=-68<BR>\n" );
document.write( "*divide by -68*<BR>\n" );
document.write( "[((x+3)^2)/7] + [((y+2)^2)/-68]=1\r<BR>\n" );
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document.write( "I think it's an ellipse, but I'm not sure. The -68 is really throwing me off.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #271739 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=271739\">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( "The -68 should throw you off.  That's because your arithmetic is faulty.  And your arithmetic is faulty because you are not following the standard process.\r<BR>\n" );
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document.write( "The first thing is to make the identification of the type of conic.\r<BR>\n" );
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document.write( "If you have an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2\"> term but no <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y^2\"> term (or vice versa), then you have a parabola.\r<BR>\n" );
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document.write( "If you have opposite 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, then you have a hyperbola.\r<BR>\n" );
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document.write( "If you have the same signs, but different coefficients (as in your example) you have an ellipse.  The squared variable with the smaller coefficient identifies the major axis.\r<BR>\n" );
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document.write( "Otherwise it is a circle.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ +\ y^2\ -\ 24x\ +\ 4y\ +\ 36\ =\ 0\">\r<BR>\n" );
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document.write( "Both variables squared (not parabola), same sign on squared variables (not hyperbola), different coefficients on the squared variables (not circle, therefore ellipse)\r<BR>\n" );
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document.write( "The first step is to move the constant term to the RHS and arrange like variable terms together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4x^2\ -\ 24x\ +\ y^2\ +\ 4y\ =\ -36\">\r<BR>\n" );
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document.write( "Start with the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> terms.  If there is a lead coefficient, then factor it out of the <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\ -\ 6x\ \ \ \right)\ +\ y^2\ +\ 4y\ =\ -36\">\r<BR>\n" );
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document.write( "Divide the coefficient on the first degree <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> term by 2, square the result, add that result INSIDE the parentheses, then add the same number TIMES the lead coefficient to the RHS:  (-6 divided by 2 is -3, -3 squared is 9, put 9 inside the parentheses, then add 4 times 9 = 36 to 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^2\ -\ 6x\ +\ 9\right)\ +\ y^2\ +\ 4y\ =\ -36\  +\ 36\">\r<BR>\n" );
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document.write( "Factor the perfect square in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x\ -\ 3\right)^2\ +\ y^2\ +\ 4y\ =\ -36\  +\ 36\">\r<BR>\n" );
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document.write( "No lead coefficient on the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> terms, so just divide the first degreee term by 2, square the result, add that result to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\left(x\ -\ 3\right)^2\ +\ y^2\ +\ 4y\ +\ 4\ =\ -36\  +\ 36\ +\ 4\">\r<BR>\n" );
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document.write( "Factor the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> terms and collect terms 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\ -\ 3\right)^2\ +\ \left(y\ +\ 2\right)^2\ =\ 4\">\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 \ \ \ \ \ \ \ \ \ \ \left(x\ -\ 3\right)^2\ +\ \frac{\left(y\ +\ 2\right)^2}{4}\ =\ 1\">\r<BR>\n" );
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document.write( "For the purposes of easy pattern matching I'm going to use the following equivalent equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\left(x\ -\ 3\right)^2}{1^2}\ +\ \frac{\left(y\ -\ (-2)\right)^2}{2^2}\ =\ 1\">\r<BR>\n" );
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document.write( "Consider the canonical equation for an ellipse with a vertical major 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}{b^2}\ +\ \frac{\left(y\ -\ k\right)^2}{a^2}\ =\ 1\">.\r<BR>\n" );
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document.write( "which has a center at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (h,k)\">, vertices at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (h, a\,+\,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (h,-a\,+\,k)\">, foci 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}\">, the end points of the major axis at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (b\,+\,h,k)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-b\,+\,h,k)\">.  The directrices are the lines <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ \frac{a^2}{c}\ +\ k\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ -\frac{a^2}{c}\ +\ k\">.\r<BR>\n" );
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document.write( "The only value you need to calculate from your standard form equation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> since the others are available by inspection.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ \sqrt{2^2\ -\ 1^2}\ =\ \sqrt{3}\">.  Plug in the numbers and you will have all of the answers you need.  If you write back, I'll send you a nice looking graph with all of the labels, etc.\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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