Question

Determine the appropriate rotation formulas to use so that the new equation contains no xy-term.

Answer 1

Step 1 :

Rotation formulas :

If the x and y-axes are rotated through an angle , the coordinates  of a point P relative to the xy-plane and the coordinates of the same point relative to the new x' and y'-axis and are related by the formulas and .

The general form is

The angle is .

If , then , so .

If , then , so .

Step 2 :

The equation is .

Compare with

and .

Substitute and in

Since , then .

Step 3 :

Rotation of x-axis :

.

Substitute in above equation.

Rotation of y-axis :

.

Substitute in above equation.

The rotation formulas are and .

Solution :

The rotation formulas are and .