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.

xy + 1 = 0

Answer 1

Step 1:

The equation is .

Definition of rotation of axes:
The general second-degree equation can be written as

by rotating the axes by an angle ,

where and .

Compare the equation with general equation.

Here and

Step 2:

Substitute in .

Consider .

Consider .

Substitute and  in the equation .

The rotated equation is .

Comments

Step 3:

Graph:

(1) Draw the coordinate plane.

(2) Draw the rotated coordinate plane

(3) Graph the equation .

Solution:

The rotated equation is

Graph of equation with rotated axis is