document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1102379>Question 1102379</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation for the ellipse with center <BR>\n" );
document.write( "(3, 0),<BR>\n" );
document.write( " foci <BR>\n" );
document.write( "(3, ±2)<BR>\n" );
document.write( " and major axis of length 6. \r<BR>\n" );
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document.write( "Find an equation for the ellipse with center <BR>\n" );
document.write( "(3, 0),<BR>\n" );
document.write( " foci <BR>\n" );
document.write( "(3, ±4)<BR>\n" );
document.write( " and major axis of length 10. <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717072 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=717072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An ellipse with foci (3, ±2)<BR>\n" );
document.write( "must have a center midway between the foci, at (3,0),<BR>\n" );
document.write( "and its focal distance must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2 ALIGN=MIDDLE  ALT=\"c=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the distance between center and either focus.<BR>\n" );
document.write( "It also must have a major axis on the line x=3,<BR>\n" );
document.write( "where the foci and center are located.<BR>\n" );
document.write( "If the length of the major axis is 6,<BR>\n" );
document.write( "the semi-major axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6%2F2=3 ALIGN=MIDDLE  ALT=\"a=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that tells us the vertices are 3 units above and below the center,<BR>\n" );
document.write( "at (3,-3) and (3,3).<BR>\n" );
document.write( "In an ellipse, the semi-minor axis <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Bc%5E2=a%5E2 ALIGN=MIDDLE  ALT=\"b%5E2%2Bc%5E2=a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B2%5E2=3%5E2 ALIGN=MIDDLE  ALT=\"b%5E2%2B2%5E2=3%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B4=9 ALIGN=MIDDLE  ALT=\"b%5E2%2B4=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=9-4 ALIGN=MIDDLE  ALT=\"b%5E2=9-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=5 ALIGN=MIDDLE  ALT=\"b%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation for an ellipse where center, foci, and vertices differ only on the y-coordinate is<BR>\n" );
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;\" > , where (h,k) is the center.<BR>\n" );
document.write( "Substituting the values we now know for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x-3%29%5E2%2F5%2By%5E2%2F9=1%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x-3%29%5E2%2F5%2By%5E2%2F9=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the ellipse, wit its foci looks like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C7%2C-4%2C4%2Cgrid%281%29%0D%0Ablue%28arc%283%2C0%2C2sqrt%285%29%2C6%2C0%2C360%29%29%2C%0D%0Ablue%28circle%283%2C2%2C0.1%29%29%2Cblue%28circle%283%2C-2%2C0.1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The equation for an ellipse with foci at (3, ±4) and major axis 10<BR>\n" );
document.write( "can be found the same way.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=10%2F2=5 ALIGN=MIDDLE  ALT=\"a=10%2F2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%7D%7D+and+%7B%7B%7B5 ALIGN=MIDDLE  ALT=\"4%7D%7D+and+%7B%7B%7B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > are leg and hypotenuse of a 3-4-5 right triangle,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=9 ALIGN=MIDDLE  ALT=\"b%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x-3%29%5E2%2F9%2By%5E2%2F25=1%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x-3%29%5E2%2F9%2By%5E2%2F25=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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