document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=857851>Question 857851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertices and foci of the ellipse given by the equation<BR>\n" );
document.write( "x^2/16 + y^2/64 = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #517007 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=517007\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you know basic algebra and learn the meaning of \"ellipse\" and related words, such as \"vertices\", \"foci\", and \"major axis\", you know all you need to know about ellipses.\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F16+%2B+y%5E2%2F64+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2F16+%2B+y%5E2%2F64+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you notice the following:<BR>\n" );
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document.write( "1) The ellipse is symmetrical with respect to the x- and y-axes (which means its center is (0,0), the origin).<BR>\n" );
document.write( "You know that because if the coordinates of a point (p,q) satisfy the equation,<BR>\n" );
document.write( "so will the coordinates of the symmetrical points (p,-q) and (-p,q).<BR>\n" );
document.write( " <BR>\n" );
document.write( "2) When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the first term is zero, and you get the most extreme values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=8%2C+%22or%22%2C+y=-8%29 ALIGN=MIDDLE  ALT=\"system%28y=8%2C+%22or%22%2C+y=-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those are the coordinates of the points on the ellipse that are farthest from the center, (0,8) and (0,-8).<BR>\n" );
document.write( "They are the ends of the major axis; they are called vertices,<BR>\n" );
document.write( "and their distance to the center is the semi-major axis,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=8 ALIGN=MIDDLE  ALT=\"a=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "3) When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the first term is zero, and you get the most extreme values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=4%2C+%22or%22%2C+x=-4%29 ALIGN=MIDDLE  ALT=\"system%28x=4%2C+%22or%22%2C+x=-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "They are the ends of the minor axis, sometimes called co-vertices,<BR>\n" );
document.write( "and their distance to the center is the semi-minor axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The foci are on the major axis (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , as the center and vertices),<BR>\n" );
document.write( "and their distance to the center, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , can be calculated from 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;\" > (NOTE below tells you why).<BR>\n" );
document.write( "In this case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2=4%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"8%5E2=4%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64=16%2Bc%5E2 ALIGN=MIDDLE  ALT=\"64=16%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64-16=c%5E2 ALIGN=MIDDLE  ALT=\"64-16=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48=c%5E2 ALIGN=MIDDLE  ALT=\"48=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%2848%29 ALIGN=MIDDLE  ALT=\"c=sqrt%2848%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4sqrt%283%29 ALIGN=MIDDLE  ALT=\"c=4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=about6.93 ALIGN=MIDDLE  ALT=\"c=about6.93\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That means that for the foci, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=4sqrt%283%29%2C+%22or%22%2C+y=-4sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"system%28y=4sqrt%283%29%2C+%22or%22%2C+y=-4sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The foci are at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%280%2C%22+-4sqrt%283%29 ALIGN=MIDDLE  ALT=\"%22%280%2C%22+-4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%280%2C%22 ALIGN=MIDDLE  ALT=\"%22%280%2C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4sqrt%283%29 ALIGN=MIDDLE  ALT=\"4sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22 ALIGN=MIDDLE  ALT=\"%22%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<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;\" > is not new to you.<BR>\n" );
document.write( "It is the Pythagorean relationship between the sides of a right triangle.<BR>\n" );
document.write( "If you look at the definition of ellipse:<BR>\n" );
document.write( "\"the locus of the points on a plane such that<BR>\n" );
document.write( "the sum of their distances to two fixed points called foci,<BR>\n" );
document.write( "equals a constant,\"<BR>\n" );
document.write( "and at the distances in the figure below,<BR>\n" );
document.write( "you find the right triangle involved.<BR>\n" );
document.write( "You find that the sum of distances for one of the vertices is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "so for the co-vertices, the distance to each focus should be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the hypotenuse of a right triangle with legs <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%280%29%2Cred%28arc%280%2C0%2C8%2C16%2C0%2C360%29%29%2C%0D%0Ablue%28circle%280%2C0%2C0.2%29%29%2Clocate%280.1%2C3.7%2Cblue%28c%29%29%2C%0D%0Ablue%28circle%280%2C8%2C0.2%29%29%2Cblue%28circle%280%2C-8%2C0.2%29%29%2C%0D%0Ablue%28circle%280%2C6.93%2C0.2%29%29%2Cblue%28circle%280%2C-6.93%2C0.2%29%29%2C%0D%0Ablue%28circle%284%2C0%2C0.2%29%29%2Cblue%28circle%28-4%2C0%2C0.2%29%29%2C%0D%0Ablue%28triangle%280%2C0%2C0%2C-6.93%2C4%2C0%29%29%2Cblue%28triangle%280%2C6.93%2C0%2C0%2C4%2C0%29%29%2C%0D%0Ablue%28rectangle%280%2C0%2C0.5%2C0.5%29%29%2Clocate%282.1%2C0.9%2Cblue%28b%29%29%2C%0D%0Ablue%28arrow%28-9.8%2C8%2C-0.2%2C8%29%29%2Clocate%28-9.8%2C8.9%2Cblue%28vertex%29%29%2C%0D%0Agreen%28arrow%28-6.5%2C8%2C-6.5%2C0%29%29%2Cgreen%28arrow%28-6.5%2C0%2C-6.5%2C8%29%29%2C%0D%0Agreen%28arrow%28-0.2%2C-8%2C-12%2C-8%29%29%2Clocate%28-6.6%2C4.5%2Ca%29%2Clocate%28-6.6%2C-3.5%2Ca%29%2C%0D%0Agreen%28arrow%28-6.5%2C-8%2C-6.5%2C0%29%29%2Cgreen%28arrow%28-6.5%2C0%2C-6.5%2C-8%29%29%2C%0D%0Alocate%282.1%2C4.3%2Cblue%28a%29%29%2Clocate%280.1%2C-2.5%2Cblue%28c%29%29%2C%0D%0Ablue%28arrow%286%2C6.93%2C0.2%2C6.93%29%29%2Clocate%286.1%2C7.4%2Cblue%28focus%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );