document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1045442>Question 1045442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The orbit of a planet around a star is described by the equation (x^2/640,000)+(y^2/630,000) = 1 ,  wherein star is at one focus, and all units are in millions of kilometers. The planet is closest and farthest from the star, when it is at the vertices. How far is the planet when it is farthest from the star? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #660878 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=660878\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2F640000%29%2B%28y%5E2%2F630000%29+=+1 ALIGN=MIDDLE  ALT=\"%28x%5E2%2F640000%29%2B%28y%5E2%2F630000%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solution: \r<BR>\n" );
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document.write( "The ellipse has center at the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=origin ALIGN=MIDDLE  ALT=\"origin\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=major ALIGN=MIDDLE  ALT=\"major\"   DISABLED_style= \"max-width:90%; height:auto;\" > axis on the x-axis.\r<BR>\n" );
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document.write( "if you compare  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2F640000%29%2B%28y%5E2%2F630000%29+=+1 ALIGN=MIDDLE  ALT=\"%28x%5E2%2F640000%29%2B%28y%5E2%2F630000%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2Fa%5E2%29%2B%28y%5E2%2Fb%5E2%29+=+1 ALIGN=MIDDLE  ALT=\"%28x%5E2%2Fa%5E2%29%2B%28y%5E2%2Fb%5E2%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you see that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2+=+640000 ALIGN=MIDDLE  ALT=\"+a%5E2+=+640000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+800 ALIGN=MIDDLE  ALT=\"a+=+800\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the vertices are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5D ALIGN=MIDDLE  ALT=\"V%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-800 ALIGN=MIDDLE  ALT=\"-800\"   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=V%5B2%5D ALIGN=MIDDLE  ALT=\"V%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=800 ALIGN=MIDDLE  ALT=\"800\"   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;\" >). \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2+=+640000 ALIGN=MIDDLE  ALT=\"+a%5E2+=+640000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+630000 ALIGN=MIDDLE  ALT=\"b%5E2+=+630000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=sqrt%28a%5E2-b%5E2%29+ ALIGN=MIDDLE  ALT=\"c+=sqrt%28a%5E2-b%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=sqrt%28640000-630000%29+ ALIGN=MIDDLE  ALT=\"c+=sqrt%28640000-630000%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=sqrt%2810000%29 ALIGN=MIDDLE  ALT=\"c+=sqrt%2810000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=100 ALIGN=MIDDLE  ALT=\"c=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Suppose the star is at the focus at the right of the origin (this choice is arbitrary, since we could have chosen instead the focus on the left). Its location is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" >(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   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;\" >). <BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=closest ALIGN=MIDDLE  ALT=\"closest\"   DISABLED_style= \"max-width:90%; height:auto;\" > distance is then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B2%5DF+=+700 ALIGN=MIDDLE  ALT=\"V%5B2%5DF+=+700\"   DISABLED_style= \"max-width:90%; height:auto;\" > (million kilometers) and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=farthest ALIGN=MIDDLE  ALT=\"farthest\"   DISABLED_style= \"max-width:90%; height:auto;\" > distance is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%5B1%5DF+=+900 ALIGN=MIDDLE  ALT=\"V%5B1%5DF+=+900\"   DISABLED_style= \"max-width:90%; height:auto;\" > (million kilometers).\r<BR>\n" );
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document.write( "Answer: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=700 ALIGN=MIDDLE  ALT=\"700\"   DISABLED_style= \"max-width:90%; height:auto;\" > million km,the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=closest ALIGN=MIDDLE  ALT=\"closest\"   DISABLED_style= \"max-width:90%; height:auto;\" > distance<BR>\n" );
document.write( "and  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900 ALIGN=MIDDLE  ALT=\"900\"   DISABLED_style= \"max-width:90%; height:auto;\" > million km, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=farthest ALIGN=MIDDLE  ALT=\"farthest\"   DISABLED_style= \"max-width:90%; height:auto;\" > distance\r<BR>\n" );
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document.write( " </SPAN>\n" );
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