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.

9x^2 + 24xy + 16y^2 + 90x - 130y = 0

Answer 1

Step 1:

Rotation formula :

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

Since . the angle lies in second quadrant.

Step 3:

Rotation of x-axis :

.

Substitute  in above equation.

Rotation of y-axis :

.

Substitute in above equation.

The rotation formulas are  and .

Answer 2

Contd...

Step 4 :

Substitute and in .

Complete the square.

This equation is the standard form of the parabola.

Step 5:

(1) Draw the coordinate plane.

(2) Draw the rotated coordinate plane.

The graph of the function .

Solution :

The angle is

The function .

The graph of the function :