document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1136451>Question 1136451</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the polar equation of an ellipse with a focus at the pole and major axis endpoints (4, 0)<BR>\n" );
document.write( "and (2, pi). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #754223 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=754223\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> given: ellipse with a focus at the pole and major axis endpoints (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "substitute in points:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=+ke%2F+%28+1+-+cos+%280%29+%29 ALIGN=MIDDLE  ALT=\"4+=+ke%2F+%28+1+-+cos+%280%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+%280%29=1 ALIGN=MIDDLE  ALT=\"cos+%280%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=+ke%2F+%28+1+-+e%281%29+%29 ALIGN=MIDDLE  ALT=\"4+=+ke%2F+%28+1+-+e%281%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+=+ke+%2F+%28+1+-+e+%29+ ALIGN=MIDDLE  ALT=\"4+=+ke+%2F+%28+1+-+e+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ke=4%28+1+-+e+%29+ ALIGN=MIDDLE  ALT=\"ke=4%28+1+-+e+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ke=4+-+4e ALIGN=MIDDLE  ALT=\"ke=4+-+4e\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4%2Fe-4 ALIGN=MIDDLE  ALT=\"k=4%2Fe-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >......eq.1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+ke%2F+%28+1+-cos+%28pi%29%29 ALIGN=MIDDLE  ALT=\"2+=+ke%2F+%28+1+-cos+%28pi%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos+%28pi%29=-1 ALIGN=MIDDLE  ALT=\"cos+%28pi%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+ke%2F+%28+1+-+e%28-1%29+%29 ALIGN=MIDDLE  ALT=\"2+=+ke%2F+%28+1+-+e%28-1%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+ke%2F+%28+1+%2B+e+%29 ALIGN=MIDDLE  ALT=\"2+=+ke%2F+%28+1+%2B+e+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ke=2+%2B+2e+ ALIGN=MIDDLE  ALT=\"ke=2+%2B+2e+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2%2Fe%2B2 ALIGN=MIDDLE  ALT=\"k=2%2Fe%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >......eq.2\r<BR>\n" );
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document.write( "from eq.1 and eq.2 we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fe-4=2%2Fe%2B2 ALIGN=MIDDLE  ALT=\"4%2Fe-4=2%2Fe%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fe-2%2Fe=2%2B4 ALIGN=MIDDLE  ALT=\"4%2Fe-2%2Fe=2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fe=6 ALIGN=MIDDLE  ALT=\"2%2Fe=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=6%2Ae ALIGN=MIDDLE  ALT=\"2=6%2Ae\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=2%2F6 ALIGN=MIDDLE  ALT=\"e=2%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e=1%2F3 ALIGN=MIDDLE  ALT=\"e=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4%2Fe-4 ALIGN=MIDDLE  ALT=\"k=4%2Fe-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >......eq.1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4%2F%281%2F3%29-4 ALIGN=MIDDLE  ALT=\"k=4%2F%281%2F3%29-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=12-4 ALIGN=MIDDLE  ALT=\"k=12-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=8 ALIGN=MIDDLE  ALT=\"k=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=+ke+%2F+%28+1+-+e+%2Acos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >...substitute in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=%288%281%2F3%29%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=%288%281%2F3%29%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=%288%2F3%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=%288%2F3%29+%2F+%28+1+-+%281%2F3%29+%2Acos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >......both numerator and denominator multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=3%288%2F3%29+%2F+%28+3+-+3%281%2F3%29+%2Acos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=3%288%2F3%29+%2F+%28+3+-+3%281%2F3%29+%2Acos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=8+%2F+%28+3+-+cos+%28theta%29%29 ALIGN=MIDDLE  ALT=\"r+=8+%2F+%28+3+-+cos+%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );