document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=311310>Question 311310</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write an equation of an ellipse centered at the origin that has a horozontal axis of length 20 and vertical axis of length 8.\r<BR>\n" );
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document.write( "Jeff said a circle is also an Ellipse. Is he correct. Explain.\r<BR>\n" );
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document.write( "Find the center Major axis, minor axis and the ellipse with equation<BR>\n" );
document.write( "(x+3)^2/9 + y^2/25 =1\r<BR>\n" );
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document.write( "Please please help me. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #222658 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=222658\">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( "General equation of a horizontally-oriented ellipse with center at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(h,k\right)\">, semi-major axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and semi-minor axis <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ >\ b\">:\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)^2}{b^2}\ =\ 1\">\r<BR>\n" );
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document.write( "So if the major axis measures 20, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 10\">, if the minor axis measures 8, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ 4\">, and if it is centered at the origin, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large h\ =\ 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\ =\ 0\">, therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x,y)\ =\ \frac{\left(x\,-\,0\right)^2}{10^2}\ +\ \frac{\left(y\,-\,0\right)^2}{4^2}\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x,y)\ =\ \frac{x^2}{100}\ +\ \frac{y^2}{16}\ =\ 1\">\r<BR>\n" );
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document.write( "==============================================================================\r<BR>\n" );
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document.write( "A circle is just a special case of an ellipse with the following three characteristics:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">The measure of the major axis is equal to the measure of the minor axis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">The two foci of the ellipse are coincident, and that point is called the center of the circle.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">The eccentricity, which is the ratio of the distance between the foci and the measure of the major axis, is zero.\r<BR>\n" );
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document.write( "All of that makes perfectly good sense algebraically.  Consider the general equation of an ellipse:\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)^2}{b^2}\ =\ 1\">\r<BR>\n" );
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document.write( "Now consider this ellipse where the measures of the major and minor axes are equal, which is to say:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ b\">.  In this case, we can re-write the general equation thus:\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)^2}{a^2}\ =\ 1\">\r<BR>\n" );
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document.write( "And then multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a^2\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(x\,-\,h\right)^2\ +\ \left(y\,-\,k\right)^2\ =\ a^2\">\r<BR>\n" );
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document.write( "If you allow <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> to represent the radius of the circle, since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> is one-half of either of the identical axes of this special ellipse, and that makes either of the axes be the diameter of the circle hence anything that is half of the diameter is by definition a radius, then you can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(x\,-\,h\right)^2\ +\ \left(y\,-\,k\right)^2\ =\ r^2\">\r<BR>\n" );
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document.write( "Which is exactly the general equation of a circle centered at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(h,k\right)\"> with radius <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\">.\r<BR>\n" );
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document.write( "==============================================================================\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}{9} + \frac{y^2}{25}\ =\ 1 \">\r<BR>\n" );
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document.write( "Re-write to fit the pattern of the general form:\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}{9} + \frac{\left(y\,-\,0\right)^2}{25}\ =\ 1 \">\r<BR>\n" );
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document.write( "Note that for this ellipse <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ <\ b\">.  That means that the ellipse is oriented vertically.  So, by examination:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h\ =\ -3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k\ =\ 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">Center: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-3,0\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a\ =\ 3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ b\ =\ 5\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">Minor Axis: 6\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \">Major Axis: 10\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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