Exploring Analyic Geometry with Mathematica®

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Center of a Quadratic

center.html

Exploration

Show that applying the change in variables

"center_1.gif" and "center_2.gif"

to the quadratic equation  "center_3.gif" causes the linear terms to vanish, implying that the center of the conic is

"center_4.gif".

Approach

Directly apply the change in variables to the equation and simplify the resulting quadratic.

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

Apply the specified change in variables.

Clear[a,b,c,d,e,f,x,y];
eq1=a*x^2+b*x*y+c*y^2+d*x+e*y+f /.
      {x->x+(2c*d-b*e)/(b^2-4a*c),
       y->y+(2a*e-b*d)/(b^2-4a*c)}

"center_5.gif"

Simplify the quadratic and notice that the linear terms have vanished.

Q1=Quadratic2D[eq1,{x,y}] //FullSimplify

"center_6.gif"

Discussion

Notice that the coefficients a, b and c are unaffected by this change in variables.


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