# SOLUTION: I have a test tomorrow that I really hope to do well on. But I seriously do not understand this chapter what so ever. How can you indentify the difference between ellipses parabola

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 Algebra: Conic sections - ellipse, parabola, hyperbola Solvers Lessons Answers archive Quiz In Depth

 Question 603025: I have a test tomorrow that I really hope to do well on. But I seriously do not understand this chapter what so ever. How can you indentify the difference between ellipses parabolas hyperbolas and circles in the equation form? Can you please help me identify and graph them, I seem to understand how to do perfect squares. I don't understand how to find the foci asymptotes latus rectum.Answer by AnlytcPhil(1277)   (Show Source): You can put this solution on YOUR website!```All of them have the form of 5 terms: Ax² + Cy² + Dx + Ey + F = 0 Get them in this form with 0 on the right. If there are fewer than 5 terms, then put 0 terms for the missing ones. For example if you have 2y² + x - 2 = 0 then consider that as the same as 0x² + 2y² + 1x + 0y - 2 = 0 and then A=0, C=2, D=1, E=0, and F=-2 [Don't confuse the capital "A" and "C" with the little "a" and "c" in the standard forms of the ellipses and hyperbolas.] ------ To determine which conic section the graph is of, If A=C then it is a circle. If A or C is 0 then it is a parabola. If A and C have the same sign but not equal it is an ellipse. If A and C have opposite signs then it is a hyperbola. 1. If A=C then it is a CIRCLE and can be placed in the form (x-h)² + (y-k)² = r² with center (h,k) and radius r 2. If A and C have the same sign but are not equal, then it is an ELLIPSE and can either be placed in this form: + = 1 which looks like this: or it is an ELLIPSE and can either be placed in this form: + = 1 which looks like this: where the center is (h,k), "a" is the length of the red line and "b" is the length of the green line. "a" is always longer than "b". You can tell which of the two forms it is by observing whether the larger which is a² is under the term (x-h)² or under the term (y-k)². The foci, or focal points, are the two points on the red line which are "c" units from the center. "c" is calculated by c² = a²-b² for all ellipses. --- 3. If A and C have opposite signs, then it is a HYPERBOLA and can either be placed in this form: - = 1 which looks like this: or it is a HYPERBOLA and can either be placed in this form: - = 1 which looks like this: where the center is (h,k), "a" is the length of the red line and "b" is the length of the green line. [CAUTION: you can't go by the length of "a" and "b" in a hyperbola as you can with an ellipse. Sometimes "a" is larger than "b", sometimes b is larger than a, and sometimes they are equal.] You can tell which of the two forms it is by observing that a² is always under the POSITIVE term and b² is always under the negative term. The foci, or focal points, are the two points on the extended red line which are "c" units from the center. "c" is calculated by c² = a²+b² for all hyperbolas. Notice that there is plus sign, whereas the formula for c in the ellipse there is a minus sign. The asymptotes are the extended diagonal of the defining rectangle. ------------------------------- If A=0, then it can be placed in the form (x - h)² = 4p(y - k) and looks like one of these: OR The vertex is the point (h,k). The red and green lines are both "|p|" units long. The blue line is the latus rectum and it is "|4p|" units long. The focal point, or focus, is the midpoint of the latus rectum. The black line is the directrix. The parabola opens upward as in the first graph if p is positive and downward if p is negative. ----------------------------- If C=0, then it can be placed in the form (y - k)² = 4p(x - h) and looks like one of these: OR The vertex is the point (h,k). The red and green lines are both "|p|" units long. The blue line is the latus rectum and it is "|4p|" units long. The focal point, or focus, is the midpoint of the latus rectum. The black line is the directrix. The parabola opens RIGHTward as in the first graph if p is positive and LEFTward if p is negative. There are also a lot of graphs of these on this site. There are also some good videos on youtube http://www.youtube.com/watch?v=5nxT6LQhXLM&feature=relmfu http://www.youtube.com/watch?v=BnLlKv6-DbA&feature=fvwrel http://www.youtube.com/watch?v=Z6cwpsDC_5A&feature=fvwrel http://www.youtube.com/watch?v=-1MzoyzWxo4&feature=related http://www.youtube.com/watch?v=04nBaKx9wiM&feature=related http://www.youtube.com/watch?v=kRT7quN7uBU&feature=related http://www.youtube.com/watch?v=k7wSPisQQYs&feature=related Edwin```