document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=605622>Question 605622</A>: <I><SPAN STYLE=\"word-break: break-all;\"> locate the center, foci, vertices, and ends of the latera recta of the ellipse. find the equation of the ellipse satisfying the given conditions. a focus at (-3,-1), one end of the minor axis at (0,3), major axis vertical </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #381770 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=381770\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The major axis is vertical and passes through focus (-3,-1), so it is the line x=-3.<BR>\n" );
document.write( "The minor axis has to be horizontal, and since it ends at (0,3), it must be the line y=3.<BR>\n" );
document.write( "Now I know that the center of the ellipse is at (-3,3), where the minor and major axes intercept.<BR>\n" );
document.write( "The length of the semi-minor axis is the distance between center (-3,3) and minor axis end (0,3):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=abs%28-3-0%29 ALIGN=MIDDLE  ALT=\"b=abs%28-3-0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=3%29 ALIGN=MIDDLE  ALT=\"highlight%28b=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The focal distance is the distance between center (-3,3) and focus (-3,-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=abs%283-%28-1%29%29 ALIGN=MIDDLE  ALT=\"c=abs%283-%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=4%29 ALIGN=MIDDLE  ALT=\"highlight%28c=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The length of the semi-major axis of an ellipse can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%28b%5E2%2Bc%5E2%29 ALIGN=MIDDLE  ALT=\"a=sqrt%28b%5E2%2Bc%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%283%5E2%2B4%5E2%29 ALIGN=MIDDLE  ALT=\"a=sqrt%283%5E2%2B4%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=5%29 ALIGN=MIDDLE  ALT=\"highlight%28a=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With that we can write the equation as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fb%5E2%2By%5E2%2Fa%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fb%5E2%2By%5E2%2Fa%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > under <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the major axis is parallel top the y-axis)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F3%5E2%2By%5E2%2F5%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F3%5E2%2By%5E2%2F5%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F9%2By%5E2%2F25=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F9%2By%5E2%2F25=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "The value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > also gives us the position of the vertices (the ends of the major axis), which are at a distance 5 above and below the center: (-3,8) and (-3, -2).<BR>\n" );
document.write( "The other end of the minor axis is at a distance <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;\" > from the center and on the other side, at (-6,3) and the other focus is at a distance <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;\" > from the center and on the other side, at (-3,7).<BR>\n" );
document.write( "The length of the semi-latus rectum can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l=b%5E2%2Fa ALIGN=MIDDLE  ALT=\"l=b%5E2%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > so in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=l=3%5E2%2F5=9%2F5=1%264%2F5 ALIGN=MIDDLE  ALT=\"l=3%5E2%2F5=9%2F5=1%264%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the latus rectum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Al=highlight%2818%2F5%29 ALIGN=MIDDLE  ALT=\"2%2Al=highlight%2818%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The semi-latus rectum gives us the horizontal distance from the foci to ellipse points on either side of them (the 4 ends of the latera recta, with y=-1 or y=7 and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3-9%2F5=-24%2F5=-4%264%2F5 ALIGN=MIDDLE  ALT=\"x=-3-9%2F5=-24%2F5=-4%264%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3%2B9%2F5=-6%2F5=-1%261%2F5 ALIGN=MIDDLE  ALT=\"x=-3%2B9%2F5=-6%2F5=-1%261%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We can plot all the meaningful points:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-9%2C3%2C-3%2C9%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%28-3%2C8%2C0.2%29%29%2Cblue%28circle%28-3%2C-2%2C0.2%29%29%2C%0D%0Agreen%28circle%28-3%2C7%2C0.2%29%29%2Cgreen%28circle%28-3%2C-1%2C0.2%29%29%2C%0D%0Ablue%28circle%28-6%2C3%2C0.2%29%29%2Cblue%28circle%280%2C3%2C0.2%29%29%2C%0D%0Ablue%28circle%28-24%2F5%2C7%2C0.2%29%29%2Cblue%28circle%28-6%2F5%2C7%2C0.2%29%29%2C%0D%0Ablue%28circle%28-24%2F5%2C-1%2C0.2%29%29%2Cblue%28circle%28-6%2F5%2C-1%2C0.2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then we could even graph the ellipse:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-9%2C3%2C-3%2C9%2C%28x%2B3%29%5E2%2B%28y%2B1%29%5E2%3C0.05%2C%28x%2B3%29%5E2%2B%28y-7%29%5E2%3C0.05%2C%28x%2B3%29%5E2%2F9%2B%28y-3%29%5E2%2F25%3E1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );