Eliminate Cross-Term by Rotation

elimxy1.html

Exploration

Show that by rotating a quadratic through an angle θ given by

Approach

Create a quadratic and rotate it by an angle θ. Show that the coefficient of the x y term is zero.

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

Rotate the quadratic.

Q1=Rotate2D[Q,ArcTan[b/(c-a)]/2];

Simplify the coefficient of the x y  term.

Q1[[2]] //Simplify