Date: 05/12/97 at 11:24:49 From: eileen williams Subject: Conic, parabolas Dear Dr. Math, How do you derive the standard equation for a parabola from the distance formula? I also need to know shifts, reflections, and dilations for a parabola of the standard equation. I hope you will be able to help me out. Thank you, Eileen Williams
Date: 05/12/97 at 12:50:25 From: Doctor Rob Subject: Re: Conic, parabolas The distance formula for a parabola is not familiar to me by that name. I will guess that you mean the equation you get from the definition of a parabola as the locus of points whose distance from a point (the focus) and perpendicular distance from a line (the directrix) are equal. I will continue with that assumption. Let the directrix be the line y = -a, and the focus be the point (0,a). The vertex will be the midpoint of the perpendicular from the focus to the directrix, in this case (0,0). Then the distance from a general point (x,y) to the directrix is y+a, and the distance to the focus is Sqrt[x^2 + (y-a)^2]. We set these distances equal: y + a = Sqrt[x^2 + (y-a)^2] To get rid of the radical, we square both sides of the equation: y^2 + 2*a*y + a^2 = x^2 + y^2 - 2*a*y + a^2 The terms y^2 and a^2 cancel from both sides of the equation, and we can combine the terms involving a*y to get: 4*a*y = x^2 or y = x^2/(4*a) This is one of the standard forms. To move the vertex away from (0,0) to, say, (r,s), make the substitutions x = X - r and y = Y - s, then expand and solve for Y. To expand by a factor of A in the x direction and B in the y direction, make the substitutions x = X/A and y = Y/B, then expand and solve for Y. To do both, use x = (X-r)/A, y = (Y-s)/B. To reflect in the x-axis, make the substitution y = -Y. Reflection in the y-axis leaves the parabola invariant, but the substitution is x = -X. Reflection in the 45-degree line x = y is given by exchanging x and y, so that x = Y and y = X. To make arbitrary rotations, reflections, and translations, the general form of the substitutions is: x = r*X + s*Y + t y = u*X + v*Y + w Where r, s, t, u, v, and w are arbitrary real numbers, except that you want r*v - s*u to be nonzero. Sometimes you are given an equation of degree two in two variables x and y of the form: a*x^2 + b*x*y + c*y^2 + d*x + e*y + f = 0 You can tell if it is a parabola by computing b^2 - 4*a*c. If this quantity is zero, you either have a parabola or a degenerate parabola. Not all of a, b, and c are zero (for the degree really to be 2). Then the parabola is degenerate if and only if b*d = 2*a*e if and only if b*e = 2*c*d, when you get two parallel lines with slope -2*a/b = -b/(2*c). -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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