document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=170283>Question 170283</A>: <I><SPAN STYLE=\"word-break: break-all;\"> one of my questions read,given the following info, write the standard equation for each conic.\r<BR>\n" );
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document.write( "1. Ellipse with a foci (-2,0) and (2,0) and verticies (-6,0) and(6,0) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125699 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=125699\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Remember, the standard equation for any ellipse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-h%29%5E2%29%2F%28a%5E2%29%2B%28%28y-k%29%5E2%29%2F%28b%5E2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-h%29%5E2%29%2F%28a%5E2%29%2B%28%28y-k%29%5E2%29%2F%28b%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, the foci of any ellipse will lie on the major axis. \r<BR>\n" );
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document.write( "Since EVERY point given to you has a y-coordinate of 0, this means that the major axis lies horizontally. So this tells us that the length of the major axis 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;\" > (since \"a\" corresponds with the horizontal axis)\r<BR>\n" );
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document.write( "Finding the center:\r<BR>\n" );
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document.write( "To find the center, simply average the vertices to get\r<BR>\n" );
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document.write( "x-mid: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%2B6%29%2F2=0%2F2=0 ALIGN=MIDDLE  ALT=\"%28-6%2B6%29%2F2=0%2F2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the center is 0 which means that h=0\r<BR>\n" );
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document.write( "y-mid: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%2B0%29%2F2=0%2F2=0 ALIGN=MIDDLE  ALT=\"%280%2B0%29%2F2=0%2F2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the 0-coordinate of the center is 0 which means that k=0\r<BR>\n" );
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document.write( "So the center is (0,0) which tells us that h=0 and k=0\r<BR>\n" );
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document.write( "Finding the length of the Major Axis:\r<BR>\n" );
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document.write( "Now let's find the distance from the two vertices:\r<BR>\n" );
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document.write( "Distance: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28-6-6%29%5E2%2B%280-0%29%5E2%29=sqrt%28%28-12%29%5E2%2B0%5E2%29=sqrt%28144%29=12 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-6-6%29%5E2%2B%280-0%29%5E2%29=sqrt%28%28-12%29%5E2%2B0%5E2%29=sqrt%28144%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the distance between the two vertices is 12 units (note: you can just count the units in between by use of a graph). This means that the length of the major axis is 12 units.\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=12 ALIGN=MIDDLE  ALT=\"2a=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > which tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6 ALIGN=MIDDLE  ALT=\"a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Finding the length of the Minor Axis:\r<BR>\n" );
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document.write( "To find the length of the minor axis (ie find the value of \"b\"), we first need to find the distance from the center to either focus\r<BR>\n" );
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document.write( "That distance is\r<BR>\n" );
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document.write( "Distance: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%280-2%29%5E2%2B%280-0%29%5E2%29=sqrt%28%28-2%29%5E2%2B%280%29%5E2%29=sqrt%284%29=2 ALIGN=MIDDLE  ALT=\"d=sqrt%28%280-2%29%5E2%2B%280-0%29%5E2%29=sqrt%28%28-2%29%5E2%2B%280%29%5E2%29=sqrt%284%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the distance from the center to either focus is 2 units. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2 ALIGN=MIDDLE  ALT=\"c=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now the value of \"b\" can be found through the formula\r<BR>\n" );
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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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E2=b%5E2%2B2%5E2 ALIGN=MIDDLE  ALT=\"6%5E2=b%5E2%2B2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6 ALIGN=MIDDLE  ALT=\"a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2 ALIGN=MIDDLE  ALT=\"c=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36=b%5E2%2B4 ALIGN=MIDDLE  ALT=\"36=b%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square each value.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4=b%5E2 ALIGN=MIDDLE  ALT=\"36-4=b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 4 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=32 ALIGN=MIDDLE  ALT=\"b%5E2=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract and rearrange the equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=sqrt%2832%29 ALIGN=MIDDLE  ALT=\"b=sqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides. Note: only the positive square root is considered\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"b=4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root.\r<BR>\n" );
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document.write( "So the values we've found was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6 ALIGN=MIDDLE  ALT=\"a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"b=4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-h%29%5E2%29%2F%28a%5E2%29%2B%28%28y-k%29%5E2%29%2F%28b%5E2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-h%29%5E2%29%2F%28a%5E2%29%2B%28%28y-k%29%5E2%29%2F%28b%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Go back to the general equation of an ellipse\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-0%29%5E2%29%2F%286%5E2%29%2B%28%28y-0%29%5E2%29%2F%28%284%2Asqrt%282%29%29%5E2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-0%29%5E2%29%2F%286%5E2%29%2B%28%28y-0%29%5E2%29%2F%28%284%2Asqrt%282%29%29%5E2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=0 ALIGN=MIDDLE  ALT=\"h=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6 ALIGN=MIDDLE  ALT=\"a=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"b=4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-0%29%5E2%29%2F%2836%29%2B%28%28y-0%29%5E2%29%2F%2816%2A2%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-0%29%5E2%29%2F%2836%29%2B%28%28y-0%29%5E2%29%2F%2816%2A2%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36. Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2A2 ALIGN=MIDDLE  ALT=\"16%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-0%29%5E2%29%2F%2836%29%2B%28%28y-0%29%5E2%29%2F%2832%29=1 ALIGN=MIDDLE  ALT=\"%28%28x-0%29%5E2%29%2F%2836%29%2B%28%28y-0%29%5E2%29%2F%2832%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the equation of the ellipse with foci of (-2,0) and (2,0) and vertices (-6,0) and(6,0) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%2F%2836%29%2B%28y%5E2%29%2F%2832%29=1 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%2F%2836%29%2B%28y%5E2%29%2F%2832%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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