document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=434963>Question 434963</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i need help with understanding ellipse and identifying the center, vertices, and covertices, and foci... i was absent two weeks from school.. im way lost </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #301157 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20855)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=301157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "    The major axis is the segment that contains both foci and has its endpoints on the ellipse. These endpoints are called the vertices. The midpoint of major axis is the center of the ellipse. \r<BR>\n" );
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document.write( "    The minor axis is perpendicular to the major axis at the center, and the endpoints of the minor axis are called co-vertices. \r<BR>\n" );
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document.write( "    The vertices are at the intersection of the major axis and the ellipse. \r<BR>\n" );
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document.write( "    The co-vertices are at the intersection of the minor axis and the ellipse. \r<BR>\n" );
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document.write( "The general form for the standard form equation of an ellipse is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fa%5E2+%2B+y%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fa%5E2+%2B+y%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, b&#8805;a.........Horizontal Major Axis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fb%5E2+%2B+y%5E2%2Fa%5E2+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fb%5E2+%2B+y%5E2%2Fa%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, b&#8805;a..........Vertical Major Axis \r<BR>\n" );
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document.write( "For a wider-than-tall ellipse with center at (h, k), having vertices a units to either side of the center and foci c units to either side of the center, the ellipse equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fb%5E2+%2B+%28y-k%29%5E2%2Fa%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fb%5E2+%2B+%28y-k%29%5E2%2Fa%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, b&#8805;a\r<BR>\n" );
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document.write( "For a taller-than-wide ellipse with center at (h, k), having vertices a units above and below the center and foci c units above and below the center, the ellipse equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-h%29%5E2%2Fb%5E2+%2B+%28x-k%29%5E2%2Fa%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28y-h%29%5E2%2Fb%5E2+%2B+%28x-k%29%5E2%2Fa%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "An ellipse equation, in conics form, is always \"=1\". Note that, in both equations above, the h always stayed with the x and the k always stayed with the y. The only thing that changed between the two equations was the placement of the a2 and the b2. The a2 always goes with the variable whose axis parallels the wider direction of the ellipse; the b2 always goes with the variable whose axis parallels the narrower direction. Looking at the equations the other way, the larger denominator always gives you the value of a2, the smaller denominator always gives you the value of b2, and the two denominators together allow you to find the value of c2 and the orientation of the ellipse.\r<BR>\n" );
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document.write( "c = &#8730;(a²-b²) \r<BR>\n" );
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document.write( "center (h, k) <BR>\n" );
document.write( "vertices (h, k±a)<BR>\n" );
document.write( "co-vertices (h±b, k) <BR>\n" );
document.write( "foci (h, k±c) \r<BR>\n" );
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document.write( " </SPAN>\n" );
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