Question

Rotate the axes to eliminate the xy-term in the  equation. Then  write  the  equation  in  standard  form. Sketch the graph of the resulting equation, showing both sets of axes.

5x^2 - 6xy + 5y^2 - 12 = 0

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 .

The angle is .

Substitute and in

Step 3 :

Rotation of x - axis :

.

Substitute  in above equation.

Rotation of y - axis :

.

Substitute  in above equation.

The rotation formulas are  and ..

Step 4 :

Substitute and in .

Divide each side 12.

The above equation is an ellipse.

Step 5 :

Graph

(1) Draw the coordinate plane.

(2) Draw the rotated coordinate plane

(3) Graph the function .

Solution :

The equation of an ellipse is .

The graph of the function .