Exploring Analyic Geometry with Mathematica®

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Parabola Determinant

pbdet.html

Exploration

Show that the determinant

"pbdet_1.gif"=0

represents a parabola "pbdet_2.gif" passing through the points "pbdet_3.gif", "pbdet_4.gif" and "pbdet_5.gif".

Approach

Expand the determinant. Convert it to a quadratic and show that the three points satisfy 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

Expand the determinant and form a quadratic.

Clear[x,y,x1,y1,x2,y2,x3,y3];
eq1=Det[{{y,x^2,x,1},
         {y1,x1^2,x1,1},
         {y2,x2^2,x2,1},
         {y3,x3^2,x3,1}}];
q1=Quadratic2D[eq1,{x,y}]

"pbdet_6.gif"

Form an equation of the quadratic.

poly1=Polynomial2D[q1,{x,y}]

"pbdet_7.gif"

Check if each of the points is on the quadratic.

Map[(poly1 /. #)&, {{x->x1,y->y1},
                    {x->x2,y->y2},
                    {x->x3,y->y3}}] //Simplify

"pbdet_8.gif"

Discussion

This is a plot of a numerical example.

p1=Point2D[{x1,y1}];
p2=Point2D[{x2,y2}];
p3=Point2D[{x3,y3}];
Sketch2D[{p1,p2,p3,q1} //. {
   x1->1, y1->1, x2->6, y2->-1,
   x3->4, y3->2}]

"pbdet_9.gif"

Graphics saved as "pbdet01.eps".


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