Consider the quadratic equation x ² + y ² + 2x - 4y = 4. Let us put this equation into standard form by completing the square, and then determine what type of conic section we are dealing with.

This is a circle with center at (-1, 2) and radius of 3.

Notice that this quadratic equation is of the form A x ² + C y ² + D x + E y + F = 0, and it is fairly easy to determine what type of conic section we are working with.

Now consider the quadratic equation x ² + 4xy + y ² - 3x = 6. What type of a conic section will this equation produce? What do we do with the 4xy term? How does this affect the graph of the conic section? I will now answer these questions starting with the last one first.

How does the Bxy term affect the graph of the conic section? It causes the conic section to be rotated a specific angle a from the positive x-axis. What do we do with the Bxy term? We can figure out the angle of rotation and transform the equation that contains the Bxy term in it to one that is in standard position and does not contain the Bxy term anymore. What type of conic section will this equation produce? We can determine the type of conic section that this equation will produce by using the discriminant, B ² - 4AC.

FACT:

With the understanding that occasional degenerate cases may arise, the quadratic curve A x ² +Bxy + C y ² + D x + E y + F = 0 is

a.

a parabola if B ² - 4AC = 0,

b.

an ellipse if B ² - 4AC < 0,

c.

a hyperbola if B ² - 4AC > 0.

What are the degenerate cases that are mentioned in the above fact? Consider A x ² + C y ² + D x + E y + F = 0, this equation can represent one of the following.

a.

A circle if A = C ¹ 0. (Special cases: the graph is a point or there is no graph at all.)

b.

A parabola if the above equation is a quadratic in one variable and linear in the other.

c.

An ellipse if A and C are both positive or both negative. (Special cases: circles, a single point, or no graph at all.)

d.

A hyperbola if A and C have opposite signs. (Special cases: a pair of intersecting lines.)

e.

A straight line if A and C are zero and least one of D and E are different from zero.

f.

One or two straight lines if the left-hand side of the above equation can be factored into the product of two linear factors.

So let us use the discriminant test to determine what type of conic sections we are working with.

EXAMPLE 1:

Determine the type of conic section this equation, 3x ² - 18xy + 27y ² - 5x + 7y = -4 will produce.

SOLUTION:

A = 3, B = -18, C = 27

B ² - 4AC = (-18) ² - 4(3)(27) = 324 - 324 = 0

The conic section that will be produced is a parabola.

EXAMPLE 2:

Determine the type of conic section this equation, 2x ² - y ² + 4xy - 2x + 3y = 6 will produce.

SOLUTION:

A = 2, B = 4, C = -1

B ² - 4AC = 4 ² - 4(2)(-1) = 16 + 8 = 24 > 0

The conic section that will be produced is a hyperbola.

EXAMPLE 3:

Determine the type of conic section this equation, 6x ² + 2y ² + 3xy + 17y = -2 will produce.

SOLUTION:

A = 6, B = 3, C = 2

B ² - 4AC = 3 ² - 4(6)(2) = 9 - 48 = -39 < 0

The conic section that will be produced is an ellipse.

Now let us learn how to rotate the coordinate axes to change Ax ² + Bxy + Cy ² + Dx + Ey + F = 0 to a conic section in standard position. First of all, we will have to determine the angle a that we will need to rotate the equation by. To do this we will use the following equations.

After we find a , then we will write the translating equations, and the equations are the following.

x = x' cos a - y' sin a

y = x' sin a + y' cos a

So let us do a few examples of rotating the coordinate axis.

EXAMPLE 4:

Rotate the coordinate axis so that x ² + xy + y ² = 1 will be in standard position.

SOLUTION:

First, we will have to find the angle of rotation for this conic section.

A = 1, B = 1, C = 1

Using the reference triangle for a p /4 triangle, we will now determine the translating equations. Just plug in the corresponding values for the cos a and sin a.

Now sub these equations into the original equation and simplify.

EXAMPLE 5:

Rotate the coordinate axis so that xy - y - x - 1 = 0 will be in standard position.

SOLUTION:

First, find the angle of rotation.

A = 0, B = 1, C = 0

Now determine the translating equations.

Now substitute these equations into the original equation.

This is one of the degenerate conic sections. It is a pair of intersecting lines.

EXAMPLE 6:

Rotate the coordinate axis so that the following quadratic equation will be in standard position.

SOLUTION:

First, find the angle of rotation.

Using the reference triangle for p / 6, determine the translating equations.

Now substitute these equations into the original equation.