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

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

b'=0

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}

Examine the resulting coefficients.

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

The x y term is zero.

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