Exploring Analyic Geometry with Mathematica®

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Eliminate Cross-Term by Change in Variables

elimxy2.html

Exploration

Show that applying the change in variables x'=k x+y and y'=k y-x, where

"elimxy2_1.gif"

to the equation "elimxy2_2.gif" will cause the x y term to vanish and a new quadratic with the following coefficients will be formed:

"elimxy2_3.gif"

b'=0

"elimxy2_4.gif"

d'=d k-e

e'=e k+d

f'=f.

Approach

Create a quadratic and form a quadratic equation. Apply the change in variables and examine the coefficients.

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

Create a quadratic.

Clear[a,b,c,d,e,f];
Q1=Quadratic2D[a,b,c,d,e,f];

Form the quadratic equation and apply the change in variables.

Clear[x,y,k];
eq1=Equation2D[Q1,{x,y}] /.
    {x->k*x+y,y->k*y-x}

"elimxy2_5.gif"

Examine the resulting coefficients.

Q2=Quadratic2D[eq1,{x,y}]

"elimxy2_6.gif"

The x y term is zero.

Q2[[2]] /. k->(c-a)/b+Sqrt[((c-a)/b)^2+1] //Simplify

"elimxy2_7.gif"


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