Parabolas and hyperbolas - how do you find them?

  • #1

Main Question or Discussion Point

Finding linear equations for initial conditions has never been a problem from me since is easy to see (x1, y1) and (x2, y2) can be plugged into the slope equation. But what if I'm looking for a parabolic or even a hyperbolic line, how do I then find its equation with the initial condition that it must have some (x,y) and (x2, y2) points?
 

Answers and Replies

  • #2
mathman
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The general form for conic sections (parabolas, hyperbolas, ellipses, circles, two lines) looks like ax2+bxy+cy2+mx+ny+k=0. Since you can divide by any constant, there are five parameters. Therefore you need to specify the curve at five points.
 
  • #3
Well, given that I still don't see a way to make sense out of that general equation since conic secs don't have an actual slope (unless one were to find its many tangents). I'm convinced that there is a way to solve for a parabolic equation given two sets of coordinates though
 
  • #4
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You plug the coordinates into the equation for each of the 5 points. That gives you 5 equations and 5 unknowns, which you can solve for.

Eg: suppose the curve passes through the following points (im just making these points up):
(1,2); (0,0); (3,4); (-1,3); (-2,-3).

Plugging each point into the equation
[tex]x^2+b \cdot x \cdot y+c \cdot y^2+m \cdot x+n \cdot y+k=0[/tex],

you get the following 5 equations:
[tex]1^2+b \cdot 1 \cdot 2+c \cdot 2^2+m \cdot 1+n \cdot 2+k=0[/tex]
[tex]0^2+b \cdot 0 \cdot 0+c \cdot 0^2+m \cdot 0+n \cdot 0+k=0[/tex]
[tex]3^2+b \cdot 3 \cdot 4+c \cdot 4^2+m \cdot 3+n \cdot 4+k=0[/tex]
[tex](-1)^2+b \cdot (-1) \cdot 3+c \cdot 3^2+m \cdot (-1)+n \cdot 3+k=0[/tex]
[tex](-2)^2+b \cdot (-2) \cdot (-3)+c \cdot (-3)^2+m \cdot (-2)+n \cdot (-3)+k=0[/tex]

Simplifying,
[tex]1+2 b+4 c+m +2 n+k=0[/tex]
[tex]k=0[/tex]
[tex]9+12 b+16 c+3 m+4 n+k=0[/tex]
[tex]1-3 b +9 c-1 m+3 n+k=0[/tex]
[tex]4+6 b+9 c-2 m-3 n+k=0[/tex]

Now you can solve these equations for b,c,m,n, and k.

----

I went ahead and solved this system of equations since it isn't much trouble, and got the following answers:
b = 116/33
c = -95/33
m = -133/11
n = 257/33
k = 0

So,
[tex]x^2+\frac{116}{33}x \cdot y-\frac{95}{33} y^2-\frac{133}{11} x+\frac{257}{33} y=0[/tex]

Plotting the resulting figure, it looks like a hyperbola, and it passes through all the points we want. Whew!
http://img520.imageshack.us/img520/8662/hyperbolapl5.png [Broken]
 
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  • #5
Sorry that you had to go through all that trouble maze. Thanks for the example
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
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Well, given that I still don't see a way to make sense out of that general equation since conic secs don't have an actual slope (unless one were to find its many tangents). I'm convinced that there is a way to solve for a parabolic equation given two sets of coordinates though
Then you are convinced wrong. y= x2 and y= 2x2- 1
both pass through the two points (1, 1) and (-1, 1).

Even given the information that the graph is a parabola, concave upward, with vertical axis, the two points (1, 1) and (-1, 1) are not enough to define a specific parabola.
 

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