Parabola Equations

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/