document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=603952>Question 603952</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you write an ellipse in standard form when you only know the foci and the covertices? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #380904 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=380904\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Knowing the foci and the covertices is just enough.<BR>\n" );
document.write( "The center of the ellipse is the midpoint of the line segment that connects the foci.<BR>\n" );
document.write( "That should also be the midpoint of the segment that connects the covertices.<BR>\n" );
document.write( "The focal distance, c, is half of the distance between the foci.<BR>\n" );
document.write( "The semi-minor axis, b, is half the distance between the covertices.<BR>\n" );
document.write( "The semi-major axis, a, is half the distance between the vertices, and can be calculated from b and c form the formula<BR>\n" );
document.write( "<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( "If the covertices have the same x-coordinate, the minor axis that connects them is vertical and the equation of the ellipse will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E4%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E4%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > with (h,k) being the center of the ellipse.<BR>\n" );
document.write( "If the covertices have the same y-coordinate, then the minor axis is horizontal, and the equation would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E4%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E4%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(If the covertices do not have a coordinate in common, the equation is more complicated, and this is not high school math any more).<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%28300%2C200%2C-6%2C6%2C-4%2C4%2C%0D%0Aline%28-5%2C0%2C5%2C0%29%2Cline%280%2C-3%2C0%2C3%29%2C%0D%0Aline%28-4%2C0%2C0%2C3%29%2C+line%280%2C3%2C4%2C0%29%2C%0D%0Arectangle%280%2C0%2C0.4%2C0.4%29%2C%0D%0Alocate%28-0.1%2C3.6%2CB%29%2C+locate%28-0.1%2C-3.1%2CY%29%2C%0D%0Alocate%285.2%2C0.3%2CA%29%2Clocate%28-5.4%2C0.3%2CX%29%2C%0D%0Alocate%283.9%2C-0.1%2CC%29%2Clocate%28-4.1%2C-0.1%2CZ%29%2C%0D%0Alocate%280.1%2C0%2CO%29%2Clocate%280.1%2C1.8%2Cb%29%2C%0D%0Aarrow%281.7%2C-1%2C0%2C-1%29%2Carrow%282.3%2C-1%2C4%2C-1%29%2C%0D%0Alocate%281.9%2C-0.7%2Cc%29%2Clocate%282.4%2C-1.7%2Ca%29%2C%0D%0Aarrow%282.2%2C-2%2C0%2C-2%29%2Carrow%282.8%2C-2%2C5%2C-2%29%2C%0D%0Alocate%282.2%2C2%2Ca%29%2Clocate%281.8%2C0.6%2Cc%29%2C%0D%0Alocate%28-4%2C-1%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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