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 .

Step 2 :

The equation is .

Compare with

and .

Substitute and in

Use Pythagorean theorem :

 

Step 3 :

Half angle formula of sine function is .

Substitute  in above equation.

Half angle formula of cosine function is .

Substitute  in above equation.

Answer 2

Contd......

Step 4 :

Rotation of x - axis :

.

Substitute and in above equation.

Rotation of y - axis :

.

Substitute and in above equation.

The rotation formulas are  and ..

Solution :

The rotation formulas are  and .