\r\n" );
document.write( "All of them have the form of 5 terms:\r\n" );
document.write( "\r\n" );
document.write( "Ax² + Cy² + Dx + Ey + F = 0\r\n" );
document.write( "\r\n" );
document.write( "Get them in this form with 0 on the right. If there are \r\n" );
document.write( "fewer than 5 terms, then put 0 terms for the missing ones.\r\n" );
document.write( "For example if you have \r\n" );
document.write( "\r\n" );
document.write( "2y² + x - 2 = 0\r\n" );
document.write( "\r\n" );
document.write( "then consider that as the same as\r\n" );
document.write( "\r\n" );
document.write( "0x² + 2y² + 1x + 0y - 2 = 0\r\n" );
document.write( "\r\n" );
document.write( "and then\r\n" );
document.write( "\r\n" );
document.write( "A=0, C=2, D=1, E=0, and F=-2\r\n" );
document.write( "\r\n" );
document.write( "[Don't confuse the capital \"A\" and \"C\" with the little \r\n" );
document.write( "\"a\" and \"c\" in the standard forms of the ellipses and \r\n" );
document.write( "hyperbolas.]\r\n" );
document.write( "\r\n" );
document.write( "------\r\n" );
document.write( "\r\n" );
document.write( "To determine which conic section the graph is of,\r\n" );
document.write( "\r\n" );
document.write( "If A=C then it is a circle.\r\n" );
document.write( "If A or C is 0 then it is a parabola.\r\n" );
document.write( "If A and C have the same sign but not equal it is an ellipse.\r\n" );
document.write( "If A and C have opposite signs then it is a hyperbola.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1. If A=C then it is a CIRCLE and can be placed in the form\r\n" );
document.write( "\r\n" );
document.write( "(x-h)² + (y-k)² = r² with center (h,k) and radius r\r\n" );
document.write( "\r\n" );
document.write( "2. If A and C have the same sign but are not equal, then it\r\n" );
document.write( "\r\n" );
document.write( "is an ELLIPSE and can either be placed in this form:\r\n" );
document.write( "\r\n" );
document.write( "
+ 
= 1\r\n" );
document.write( "\r\n" );
document.write( "which looks like this: \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "or it is an ELLIPSE and can either be placed in this form: \r\n" );
document.write( "\r\n" );
document.write( "
+ 
= 1\r\n" );
document.write( "\r\n" );
document.write( "which looks like this:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "where the center is (h,k), \"a\" is the length of the red line \r\n" );
document.write( "and \"b\" is the length of the green line. \"a\" is always longer\r\n" );
document.write( "than \"b\". You can tell which of the two forms it is by observing\r\n" );
document.write( "whether the larger which is a² is under the term (x-h)² or under\r\n" );
document.write( "the term (y-k)².\r\n" );
document.write( "\r\n" );
document.write( "The foci, or focal points, are the two points on the red line which\r\n" );
document.write( "are \"c\" units from the center. \"c\" is calculated by c² = a²-b² for\r\n" );
document.write( "all ellipses.\r\n" );
document.write( "\r\n" );
document.write( "---\r\n" );
document.write( "3. If A and C have opposite signs, then it\r\n" );
document.write( "\r\n" );
document.write( "is a HYPERBOLA and can either be placed in this form:\r\n" );
document.write( "\r\n" );
document.write( " - = 1\r\n" );
document.write( "\r\n" );
document.write( "which looks like this:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "or it is a HYPERBOLA and can either be placed in this form: \r\n" );
document.write( "\r\n" );
document.write( " - = 1\r\n" );
document.write( "\r\n" );
document.write( "which looks like this:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "where the center is (h,k), \"a\" is the length of the red line \r\n" );
document.write( "and \"b\" is the length of the green line. \r\n" );
document.write( "\r\n" );
document.write( "[CAUTION: you can't go\r\n" );
document.write( "by the length of \"a\" and \"b\" in a hyperbola as you can with an\r\n" );
document.write( "ellipse. Sometimes \"a\" is larger than \"b\", sometimes b is larger\r\n" );
document.write( "than a, and sometimes they are equal.] \r\n" );
document.write( "\r\n" );
document.write( "You can tell which of the two forms it is by observing that a² is\r\n" );
document.write( "always under the POSITIVE term and b² is always under the negative \r\n" );
document.write( "term.\r\n" );
document.write( "\r\n" );
document.write( "The foci, or focal points, are the two points on the extended red \r\n" );
document.write( "line which are \"c\" units from the center. \"c\" is calculated by \r\n" );
document.write( "c² = a²+b² for all hyperbolas. Notice that there is plus sign,\r\n" );
document.write( "whereas the formula for c in the ellipse there is a minus sign.\r\n" );
document.write( "\r\n" );
document.write( "The asymptotes are the extended diagonal of the defining rectangle.\r\n" );
document.write( "\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "\r\n" );
document.write( "If A=0, then it can be placed in the form\r\n" );
document.write( "\r\n" );
document.write( "(x - h)² = 4p(y - k)\r\n" );
document.write( "\r\n" );
document.write( "and looks like one of these:\r\n" );
document.write( "\r\n" );
document.write( "
OR 
\r\n" );
document.write( "\r\n" );
document.write( "The vertex is the point (h,k). The red and green lines are both\r\n" );
document.write( "\"|p|\" units long. The blue line is the latus rectum and it is \"|4p|\" \r\n" );
document.write( "units long. The focal point, or focus, is the midpoint of the \r\n" );
document.write( "latus rectum. The black line is the directrix. The parabola opens\r\n" );
document.write( "upward as in the first graph if p is positive and downward if p is\r\n" );
document.write( "negative.\r\n" );
document.write( "\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "If C=0, then it can be placed in the form\r\n" );
document.write( "\r\n" );
document.write( "(y - k)² = 4p(x - h)\r\n" );
document.write( "\r\n" );
document.write( "and looks like one of these:\r\n" );
document.write( "\r\n" );
document.write( "
OR 
\r\n" );
document.write( "\r\n" );
document.write( "The vertex is the point (h,k). The red and green lines are both\r\n" );
document.write( "\"|p|\" units long. The blue line is the latus rectum and it is \"|4p|\" \r\n" );
document.write( "units long. The focal point, or focus, is the midpoint of the \r\n" );
document.write( "latus rectum. The black line is the directrix. The parabola opens\r\n" );
document.write( "RIGHTward as in the first graph if p is positive and LEFTward if p is\r\n" );
document.write( "negative.\r\n" );
document.write( "\r\n" );
document.write( "There are also a lot of graphs of these on this site. There are also\r\n" );
document.write( "some good videos on youtube\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "http://www.youtube.com/watch?v=5nxT6LQhXLM&feature=relmfu\r\n" );
document.write( "http://www.youtube.com/watch?v=BnLlKv6-DbA&feature=fvwrel\r\n" );
document.write( "http://www.youtube.com/watch?v=Z6cwpsDC_5A&feature=fvwrel\r\n" );
document.write( "\r\n" );
document.write( "http://www.youtube.com/watch?v=-1MzoyzWxo4&feature=related\r\n" );
document.write( "http://www.youtube.com/watch?v=04nBaKx9wiM&feature=related\r\n" );
document.write( "http://www.youtube.com/watch?v=kRT7quN7uBU&feature=related\r\n" );
document.write( "http://www.youtube.com/watch?v=k7wSPisQQYs&feature=related\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-12.gif\")
