a. determine the type of conic using the discriminant

The general equation for conic sections can be written as

Comparing the given equation with the general equation,

------- ■

The discriminant is

Therefore, the equation represents an .

b.find the angle of rotation

For the general equation of a conic, the angle of rotation about the origin is given by:

, if A = C

, if

In this case,

Substituting 21 for A, for B, and 31 for C,

Solving for ,

-------●

c.use rotation formulas to eliminate the xy-term and sketch

If the conic section with the equation:

is transformed by a rotation about the origin through the angle , then the equation in the new coordinates and has the form:

The coordinate transformation which expresses the old coordinates in terms of the new ones is given by:

Thus, putting the coordinate transformations into the original equation:

Opening up the brackets and expanding, we get:

Rearranging,

Thus, the parameters of the new equation are:

and using ■ and ●, we get

In our case, applying the rotation formula,

and the equation gets the form:

,

which is the equation of a standard ellipse.