Question

Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.

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 .

The angle is .

Substitute and in

Since , the angle lies in second quadrant.

Use Pythagorean theorem :

.

  

In second quadrant cosine function is negative.

Answer 2

Contd..............

Step 3 :

Half angle formula of sine function is .

Substitute  in above equation.

Half angle formula of cosine function is .

Substitute  in above equation.

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 .

Answer 3

Step 5 :

Substitute and in

.

 

The above equation is a parabola.

The general form of parabolic equation is .

Where is the vertex and is focus.

Compare with .

and

The vertex is .

The focus is .

Step 6 :

Graph

(1) Draw the coordinate plane.

(2) Draw the rotated coordinate plane

(3) Graph the function .

Solution :

The angle is

The function .

The vertex is .

The focus is .

The graph of the function