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Parabolas Through Pairs of Points

Date: 07/13/2002 at 15:57:14
From: Jeffrey G. Cooper
Subject: How many parabolas can pass through two given points?

Can an infinite number of parabolas pass through two points in a
plane?  Yes or no, and why?


Date: 07/14/2002 at 11:19:03
From: Doctor Douglas
Subject: Re: How many parabolas can pass through two given points?

Hi, Jeffrey,

Thanks for submitting your question to the Math Forum.

Yes, infinitely many parabolas can pass through two points in a plane.

Here's a simple example:

   Let the points be (0,0) and (1,0).  Then the family of parabolas

      y = A(x - 0)(x - 1) 
       
        = A(x^2 - x)

   will all pass through both points.  A is any real constant 
   (it should be nonzero for the equation to be a parabola).  So
   all of the following parabolas pass through the given points:

      y = x^2 - x

      y = 2x^2 - 2x

      y = -8x^2 - 8x

      y = -pi x^2 - pi x

   and so on.

Now, if the two given points don't lie on a horizontal or vertical
line, it's still possible to find many (actually an infinite number 
of) parabolas that pass through both points.  

Let's say that the given points are (x1,y1) and (x2,y2).  Then the
line joining them is 

  y - y1 = (y2 - y1)[(x - x1)/(x2 - x1)]

You can easily verify that if x=x1, then both sides must be zero and
y=y1.  And if x=x2, then the factor in brackets must be equal to one,
and y=y2.  

This line can also be written in the following form:  

  y = y1 + m(x - x1)

where 

  m = (y2 - y1)/(x2 - x1) 

is the slope of the line.

Now consider the following family of parabolas:

  y = y1 + m(x - x1) + A(x - x1)(x - x2)             

If x=x1, then the last two terms are zero, and y=y1.  If x=x2, 
the only the last term is zero and 

  y = y1 + m(x2 - x1) 

    = y1 + y2 - y1

    = y2, 

so any parabola (again, where A is nonzero) of this form will
be forced to pass through the two given points.  Moreover, there
are still more parabolas passing through the points that don't
open up directly downward or upward; these are not included in
the equation here

It takes three (noncollinear) points and a specified direction for 
the parabola axis (e.g. vertical) to specify a single parabola.

I hope this answers your question!  Please write back if you need
more explanation.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Conic Sections/Circles
High School Equations, Graphs, Translations
High School Polynomials

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