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 $\"\"$ .

\

Step 3:

\

The rotated equation is $\"\"$ .

\

General form of hyperbola is $\"\"$ , where

\

$\"\"$ is the center of the hyperbola,

\

The distance between center and vertex is $\"\"$ .

\

The distance between center and focus is $\"\"$ .

\

and $\"\"$ .

\

$\"\"$

\

Graph:

\

(1) Draw the coordinate plane.

\

(2) Draw the rotated coordinate plane

\

(3) Graph the equation $\"\"$ .

\

(4) Plot the center point $\"\"$ .

\

(5) Plot the focus points $\"\"$ and $\"\"$ .

\

(6) Plot the vertex points $\"\"$ and $\"\"$ .

\

$\"\"$

\

Solution:

\

The rotated equation is $\"\"$

\

Graph of equation $\"\"$ is

\

$\"\"$

\

\