document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=604369>Question 604369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write an equation of an ellipse with center at (0, 0), co-vertex at (-4 , 0), and focus at (0, 3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #381198 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=381198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> O(0,0) is the center,<BR>\n" );
document.write( "(-4,0), and of course, symmetrically (4,0) are the co-vertices,<BR>\n" );
document.write( " so 4 is the semi-minor axis, b.<BR>\n" );
document.write( "(0,3), and of course, symmetrically (0,-3) are the foci,<BR>\n" );
document.write( " so 3 is the focal distance, c.<BR>\n" );
document.write( "The vertices are at (0,a) and (0,-a), with a being the semi-major axis and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2=4%5E2%2B3%5E2=16%2B9=25=5%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2=4%5E2%2B3%5E2=16%2B9=25=5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <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( "and <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;\" > so <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;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%5E2%2F9%2By%5E2%2F25=1%29 ALIGN=MIDDLE  ALT=\"highlight%28x%5E2%2F9%2By%5E2%2F25=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "HOW I KNOW THAT <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "In the diagram below, O is the center of the ellipse, with axes XA and BY.<BR>\n" );
document.write( "(I did not draw the whole ellipse, just the important points).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28200%2C300%2C-4%2C4%2C-6%2C6%2C%0D%0Aline%280%2C-5%2C0%2C5%29%2Cline%28-3%2C0%2C3%2C0%29%2C%0D%0Aline%280%2C-4%2C3%2C0%29%2C+line%283%2C0%2C0%2C4%29%2C%0D%0Arectangle%280%2C0%2C0.4%2C0.4%29%2C%0D%0Alocate%283.1%2C0.3%2CB%29%2C+locate%28-3.3%2C0.3%2CY%29%2C%0D%0Alocate%28-0.2%2C5.6%2CA%29%2Clocate%28-0.2%2C-5.1%2CX%29%2C%0D%0Alocate%28-0.4%2C4.3%2CC%29%2Clocate%28-0.4%2C-3.7%2CZ%29%2C%0D%0Alocate%280.1%2C0%2CO%29%2Clocate%281.4%2C0%2Cb%29%2C%0D%0Aarrow%28-1%2C1.7%2C-1.0%29%2Carrow%28-1%2C2.3%2C-1%2C4%29%2C%0D%0Alocate%28-1.1%2C2.4%2Cc%29%2Clocate%28-2.1%2C2.8%2Ca%29%2C%0D%0Aarrow%28-2%2C2.2%2C-2%2C0%29%2Carrow%28-2%2C2.8%2C-2%2C5%29%2C%0D%0Alocate%281.8%2C2.2%2Ca%29%2Clocate%280.1%2C2%2Cc%29%2C%0D%0Alocate%28-3.5%2C-2%2Ca%5E2=b%5E2%2Bc%5E2%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > A and X are vertices; B and Y covertices; C and Z foci.<BR>\n" );
document.write( "The important segment lengths are:<BR>\n" );
document.write( "OA=OX=a (the semi-major axis)<BR>\n" );
document.write( "OB+OY=b (the semi-minor axis)<BR>\n" );
document.write( "OC=OZ=c (the focal distance<BR>\n" );
document.write( "Covertex B is one of the points of the ellipse.<BR>\n" );
document.write( "The distances from B to focus C and to focus Z are the same BZ=BC.<BR>\n" );
document.write( "Their sum BZ+BC=2BC is the same as the sum of distances to the foci for all points on the ellipse.<BR>\n" );
document.write( "Vertex A is one of the points on the ellipse.<BR>\n" );
document.write( "The sum of its distances to the foci is AC+AZ=AC+(AO+OZ)=(a-c)+(a+c)=2a<BR>\n" );
document.write( "2BC=2a --> BC=a<BR>\n" );
document.write( "In the right triangle OBC, Pythagoras theorem says that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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