ParabolaDate: 12/23/97 at 17:34:41 From: jen Subject: Parabola How can you, from an ordered pair, find the vertex, axis of symmetry, directrix, the focus, the distance from the vertex to the focus, and the latus rectum? I use the following two equations in class: (x-h)^2 = 4p(y-k) or (y-k)^2 = 4p(x-h). Also, how can you tell if your numbers go in the x squared or y squared equation? Thank you for your time. Date: 12/24/97 at 11:48:00 From: Doctor Rob Subject: Re: Parabola You can't find these things from a single ordered pair (x,y) of coordinates of a point on the curve. You need three pairs to do that, since there are three unknowns in the equation: h, k, and p. You also cannot tell which of the two equations you listed to use, even with three pairs. A fourth pair would let you figure out which of the two equations is the right one. With both equations, the vertex is at (h,k), its distance from the focus is |p|, and the length of the latus rectum is 4*|p|. With the first equation, the axis of symmetry is the line x = h, the directrix is the line y = k - p, and the focus is at (h,k+p). With the second equation, the axis of symmetry is the line y = k, the directrix is the line x = h - p, and the focus is at (h+p,k). Example: The points (-4,10), (-2,-3), and (3,7) lie on a parabola. If the first equation is used, then (-4-h)^2 = 4*p*(10-k), (-2-h)^2 = 4*p*(-3-k), (3-h)^2 = 4*p*(7-k). We can use the first equation to solve for k, and substitute in the other two equations: k = 10 - (4+h)^2/(4*p) (-2-h)^2 = 4*p*[-3 - 10 + (4+h)^2/(4*p)] = 4*p*(-13) + (4+h)^2, (3-h)^2 = 4*p*[7 - 10 + (4+h)^2/(4*p)] = 4*p*(-3) + (4*h)^2, Now we can use the second equation to solve for p, and substitute in the other equations: p = (h+3)/13, k = -(13*h^2+64*h+88)/(4*[3+h]) = -(13*h+25)/4 - 13/(4*[h+3]), 0 = 5*(34*h+11)/13. Using the last equation to solve for h, and substituting in the other equations, we get: h = -11/34, k = -6105/952, p = 7/34. This allows us to answer all the questions posed. If the second equation is used, we get instead, (10-k)^2 = 4*p*(-4-h), (-3-k)^2 = 4*p*(-2-h), (7-k)^2 = 4*p*(3-h), which gives the solution h = -13633/2856, k = -107/6, p = 119/6. Again, this allows us to answer all the questions posed. If you know five points, you can determine the conic section on which they lie by using the equation A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0, substituting in the five pairs of values (x,y), and getting five linear equations in the six unknowns A, B, C, D, E, and F. You can solve them for five of the unknowns in terms of the sixth. If not all of A, B, and C are zero, and B^2 - 4*A*C = 0, this is a parabola. You can determine the answers to all your questions for this parabola, too. It is very possible that its axis may be oblique, that is, not parallel to either coordinate axis. In fact the slope of the axis is B/(2*C) if C is nonzero. If C is zero, then B must also be zero, and then the axis is vertical, and parallel to the y-axis. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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