document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1119858>Question 1119858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The moon's orbit is an ellipse with earth as one focus. If the maximum diatance from the moon to earth is 405 500 km and the minimum distance is 363 300 km, find the equation of the ellipse in a cartesian coordinate system where earth is at the origin. Assume that the ellipse has horizontal major axis and that the minimum distance is achieved when the moon is to the right of the earth. Use 100 km as one unit. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #735488 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=735488\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "According to the description, we are to view the ellipse as having a horizontal major axis.  The standard form of the equation for such an ellipse is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In that form, the center of the ellipse as (h,k); a and b are the semi-major and semi-minor axes, and c is the distance from the center to each focus, where c^2 = a^2-b^2.<br><BR>\n" );
document.write( "The minimum distance between earth and the moon is 3633 units to the right of earth; the maximum distance between earth and the moon is 4055 units to the left of earth.  That means the length of the major axis is 3633+4055 = 7688 units; then the semi-major axis, a, is 7688/2 = 3844 units.<br><BR>\n" );
document.write( "c, the distance from the center of the ellipse to a focus (the earth), is then 3844-3633 = 211 units.<br><BR>\n" );
document.write( "Then, since the problem specifies that the earth be at the origin, the center of the ellipse is (-211,0).<br><BR>\n" );
document.write( "We have h=-211 and k=0; and we know a=3844, so we can calculate a^2 = 14776336.  Now we need to find b^2 using c^2 = a^2-b^2.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=211%5E2+=+3844%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"211%5E2+=+3844%5E2-b%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+=+3844%5E2-211%5E2+=+14731815 ALIGN=MIDDLE  ALT=\"b%5E2+=+3844%5E2-211%5E2+=+14731815\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we have all the numbers we need to write the equation:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-%28-211%29%29%5E2%2F14776336%2B%28y-0%29%5E2%2F14731815+=+1 ALIGN=MIDDLE  ALT=\"%28x-%28-211%29%29%5E2%2F14776336%2B%28y-0%29%5E2%2F14731815+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B211%29%5E2%2F14776336%2By%5E2%2F14731815+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B211%29%5E2%2F14776336%2By%5E2%2F14731815+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );