Exploring Analyic Geometry with Mathematica®

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Parabola Through Three Points

pb3pts.html

Exploration

Show that the parabola passing through the points (0,0), (a,b) and (b,a) whose axis is parallel to the x-axis has vertex (h,k) and focal length f given by

"pb3pts_1.gif" and "pb3pts_2.gif".

Furthermore, show that the quadratic representing the parabola is

"pb3pts_3.gif".

Approach

Create the equation of a parabola in standard position with variables (h,k) for the vertex point and f for the focal length. The three given points must satisfy the equation. Solve  three equations in three unknowns (h, k and f). Find the quadratic representing the equation.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Write the equation of the parabola in standard position.

Clear[x,y,h,k,f];
eq1=(y-k)^2==4f(x-h);

Solve for the constants.

Clear[a,b];
ans=Solve[Map[(eq1 /. #)&,
              {{x->0,y->0},
               {x->a,y->b},
               {x->b,y->a}}],
          {h,k,f}] //Simplify

"pb3pts_4.gif"

Form the quadratic representing the parabola.

q1=Quadratic2D[eq1 /. ans[[1]],{x,y}] //Simplify

"pb3pts_5.gif"

Multiply through by (a+b) to arrive at the desired form of the equation.

Equation2D[Map[(#*(a+b))&, q1],{x,y}]

"pb3pts_6.gif"

Discussion

This is a plot of a numerical example with a=2 and b=3.

Sketch2D[{Point2D[{0,0}],Point2D[{a,b}],
          Point2D[{b,a}],q1} /. {
   a->2, b->3}]

"pb3pts_7.gif"

Graphics saved as "pb3pts01.eps".


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