## Exploring Analyic Geometry with |
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Center of a Quadratic

center.html

Exploration

Show that applying the change in variables

to the quadratic equation causes the linear terms to vanish, implying that the center of the conic is

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)}

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

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

Discussion

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

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