document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106060>Question 1106060</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph of the conic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2%2B49y%5E2%2B192x=208 ALIGN=MIDDLE  ALT=\"16x%5E2%2B49y%5E2%2B192x=208\"   DISABLED_style= \"max-width:90%; height:auto;\" > has vertices and foci equal to ?? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #721043 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=721043\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The equation has both x^2 and y^2 terms, with the same sign and different coefficients, so the equation is of an ellipse.<br><BR>\n" );
document.write( "The standard form of the equation of 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;\" >  if the major axis is in the x direction; or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fb%5E2+%2B+%28y-k%29%5E2%2Fa%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fb%5E2+%2B+%28y-k%29%5E2%2Fa%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  if the major axis is in the y direction<br><BR>\n" );
document.write( "In both formulas, (h,k) is the center of the ellipse; and a and b are the semi-major and semi-minor axes, respectively.<br><BR>\n" );
document.write( "Two different forms of the equation are necessary, because for the ellipse the a has to be the length of the semi-major axis.<br><BR>\n" );
document.write( "Parameter c is the distance from the center of the ellipse to each focus; for an ellipse, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"c%5E2+=+a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Note that this formula for finding the value of c is why it is necessary to have a > b.<br><BR>\n" );
document.write( "To find the vertices and foci, you need to put the given equation in the standard form.  To do that, you need to complete the square in both x and y, then divide by the appropriate constant to get the right side of the equation equal to 1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2%2B49y%5E2%2B192x=208 ALIGN=MIDDLE  ALT=\"16x%5E2%2B49y%5E2%2B192x=208\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816x%5E2%2B192x%29%2B49y%5E2+=+208 ALIGN=MIDDLE  ALT=\"%2816x%5E2%2B192x%29%2B49y%5E2+=+208\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%5E2%2B12x%29%2B49y%5E2+=+208 ALIGN=MIDDLE  ALT=\"16%28x%5E2%2B12x%29%2B49y%5E2+=+208\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%5E2%2B12x%2B36%29%2B49y%5E2+=+208%2B16%2836%29+=+208%2B576+=+784 ALIGN=MIDDLE  ALT=\"16%28x%5E2%2B12x%2B36%29%2B49y%5E2+=+208%2B16%2836%29+=+208%2B576+=+784\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%2B6%29%5E2%2F784%2B49y%5E2%2F784+=+1 ALIGN=MIDDLE  ALT=\"16%28x%2B6%29%5E2%2F784%2B49y%5E2%2F784+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B6%29%5E2%2F49+%2B+y%5E2%2F16+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B6%29%5E2%2F49+%2B+y%5E2%2F16+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B6%29%5E2%2F7%5E2+%2B+%28y-0%29%5E2%2F4%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B6%29%5E2%2F7%5E2+%2B+%28y-0%29%5E2%2F4%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is in standard form:<BR>\n" );
document.write( "The center of the ellipse is (h,k) = (-6,0).<BR>\n" );
document.write( "a is 7; the semi-major axis has length 7 in the x direction, so the vertices are at (-13,0) and (1,0).<BR>\n" );
document.write( "b is 4; the semi-major axis has length 4 in the y direction.  If you need the co-vertices, they are 4 units in the positive and negative y direction from the center -- at (-6,-4) and (-6,4).<BR>\n" );
document.write( "The distance from the center to each focus is c, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2-b%5E2%29+=+sqrt%2849-16%29+=+sqrt%2833%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2-b%5E2%29+=+sqrt%2849-16%29+=+sqrt%2833%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So the foci are at (-6-sqrt(33),0) and (-6+sqrt(33),0).<BR>\n" );
document.write( " </SPAN>\n" );
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