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# 6.9: General Forms of Conic Sections

Difficulty Level: At Grade Created by: CK-12
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Kelli was reviewing for her pre-calc final on conic sections, and she was trying to memorize all of the rules for each of the shapes. She was actually doing pretty well, but was frustrated with the challenge of keeping them all straight.

Kelli found herself wishing there were a way to identify the shape early on, so she would have an idea of what she was generally sketching before she worried about specifics.

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### Guidance

Let’s examine all the equations of the conic sections we’ve studied in this chapter.

Ellipses: \begin{align*}\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are any positive numbers (the circle is the specific case when \begin{align*}a = b\end{align*}).

Parabolas: \begin{align*}y = ax^2\end{align*} or \begin{align*}x = ay^2\end{align*} where \begin{align*}a\end{align*} is any non-zero number.

Hyperbolas: \begin{align*}\frac{x^2}{a^2} - \frac{y^2}{b^2} =1\end{align*} or \begin{align*}\frac{y^2}{a^2} - \frac{x^2}{b^2} =1\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are any positive numbers.

All these equations have in common that they are degree-2 polynomials, meaning the highest exponent of any variable—or sum of exponents of products of variables—is two. So for example, here are some degree two polynomial equations in a more general form:

\begin{align*}2x^2 + 5y^2 - 3y + 4 & = 0 \\ x^2 - 3y^2 + x - y + 3 & = 0 \\ 10x^2 - y - 5 & = 0 \\ xy & = 2\end{align*}

Some of these probably already look like conic sections to you. For example, in the first equation, we can complete the square to remove the \begin{align*}-3y\end{align*} term and we will see that we have an ellipse. In the second equation we can complete the square twice to remove both the \begin{align*}x\end{align*} and \begin{align*}-y\end{align*} terms and we will have a hyperbola. This is a hyperbola, not an ellipse, because the coefficient of the \begin{align*}x^2\end{align*} and \begin{align*}y^2\end{align*} terms have opposite signs.

The third equation is a parabola since there is an \begin{align*}x^2\end{align*} term and \begin{align*}y\end{align*} term but not a \begin{align*}y^2\end{align*} term. Do you see how you can solve for \begin{align*}y\end{align*}, putting the equation in the standard form for a vertically oriented parabola?

But what about the fourth equation? Like the others, it is a degree-2 polynomial, since the exponents of the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} term sum to 2. But the fourth equation looks nothing like any of the forms for conic sections that we’ve examined so far. Nonetheless, as we saw in the last section, \begin{align*}xy =2\end{align*} appears to be a hyperbola with foci (2,2) and (-2,-2). The reason it doesn’t fit either of the standard forms for hyperbolas is because it is diagonally oriented, rather than horizontally or vertically oriented (do you see how its two foci lie on a diagonal line, rather than a horizontal or vertical line?)

In order to see how such differently-oriented conic sections fit into our standard forms, we need to rotate them so that they are either horizontally or vertically oriented.

Identifying Conics

How can we look at a degree-2 polynomial equation and determine which conic section it depicts?

\begin{align*}Ax^2 + By^2 + C xy + Dx + Ey + F=0\end{align*}

When \begin{align*}C=0\end{align*} we have already discussed how to determine which conic section the equation refers to. In summary, if \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are both positive, the conic section is an ellipse. This is also true of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are both negative, as the entire equation can be multiplied by -1 without changing the solution set. If \begin{align*}A\end{align*} and \begin{align*}B\end{align*} differ in sign, the equation is a hyperbola, and if \begin{align*}A\end{align*} or \begin{align*}B\end{align*} equals zero the equation is a parabola.

There are a few new, more general, rules I will show you that give more information about the case when \begin{align*}C \neq 0\end{align*} and hence the conic section needs to be rotated to achieve horizontal or vertical orientation.

If \begin{align*}C^2 < 4AB\end{align*}, the equation is an ellipse (note when \begin{align*}C=0\end{align*} this holds whenever \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are the same sign, which is consistent with our simpler rule stated above.)

If \begin{align*}C^2 > 4AB\end{align*}, the equation is an hyperbola (note when \begin{align*}C=0\end{align*} this holds whenever \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are the opposite sign, which is consistent with our simpler rule stated above.)

If \begin{align*}C^2 = 4AB\end{align*}, the equation is a parabola (note when \begin{align*}C=0\end{align*}, either \begin{align*}A\end{align*} or \begin{align*}B\end{align*} equals zero, which is consistent with our simpler rule stated above.)

Rotating Conics

Suppose we have an equation of degree two polynomials, such as the \begin{align*}xy = 2\end{align*} example discussed above. In order to put it into the more recognizable form of a ellipse, parabola, or a hyperbola, we need to rotate it in such a way so that rotated version has no \begin{align*}xy\end{align*} term. So we need to find an appropriate angle \begin{align*}\theta\end{align*} such that changing the \begin{align*}x-\end{align*}coordinate to \begin{align*}x' \cos(\theta)-y' \sin (\theta)\end{align*} and the \begin{align*}y-\end{align*}coordinate to \begin{align*}x' \sin (\theta) + y' \cos (\theta) \end{align*} results in an equation with no \begin{align*}xy\end{align*} term.

#### Example A

State what type of conic section is represented by the following equation: \begin{align*}5x^2+6y^2+2x-5y+xy=0\end{align*}.

Solution

This is an ellipse as both a and b are positive, also note that c < 4ab.

#### Example B

State what type of conic section is represented by the following equation: \begin{align*}x^2+3y-20xy+20=0\end{align*}.

Solution

This is a hyperbola, since c2 > 4ab (400 > 12).

#### Example C

Which conic section is described by the equation?

\begin{align*}4x^3 - 2x^2 +37 = 0\end{align*}

Solution

To identify \begin{align*}4x^3 - 2x^2 +37 = 0\end{align*} we just need to look at the signs of the squared terms:

The \begin{align*}x^2\end{align*} and \begin{align*}y^2\end{align*}coefficients have different signs so it's a hyperbola.
Additionally, the \begin{align*}y^2\end{align*} term is negative, so it has a vertical transverse

### Vocabulary

A tangent "just touches" another object, making contact at a single point or along a line; touching but not intersecting.

Ellipses are "squashed circles" described by \begin{align*}\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are any positive numbers (the circle is the specific case when \begin{align*}a = b\end{align*}).

Parabolas are U-shaped figures described by an equation of the form: \begin{align*}y = ax^2\end{align*} or \begin{align*}x = ay^2\end{align*} where \begin{align*}a\end{align*} is any non-zero number.

Hyperbolas are similar to two parabolas with vertexes together, extending infinitely away in the other direction. Hyperbolas can be calculated with: \begin{align*}\frac{x^2}{a^2} - \frac{y^2}{b^2} =1\end{align*} or \begin{align*}\frac{y^2}{a^2} - \frac{x^2}{b^2} =1\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are any positive numbers.

### Guided Practice

When identifying conic sections, the squared terms are the only factors that matter. Run the standard form of the equation you are questioning through the following series of tests on the squared terms:

1) If only 1 variable is squared, you have a parabola, stop further testing
2) If the coefficients of the squared terms have opposite signs, you have a hyperbola, stop further testing
3) If the squared terms are multiplied by the same coefficient, you have a circle, stop testing
4) If none of the above apply, you have an ellipse

1. Identify and describe the conic section represented by the following equation: \begin{align*}(x - 1)^2 + (y + 1)^2 = 9\end{align*}

2. State what type of conic section is represented by the following equation: \begin{align*}3x^2 - 5y^2 + 23 = 0\end{align*}

3. State what type of conic section is identified by the following equation: \begin{align*}-8x^2 - 18y^2 + 48x - 360y - 1800 = 0\end{align*}

4. State the and describe the conic section: \begin{align*}2x^2 - 16x + 32 +3y = 6\end{align*}

Solutions

1. To identify \begin{align*}(x - 1)^2 + (y + 1)^2 = 9:\end{align*}

Both terms are squared: not a parabola.
The coefficients of \begin{align*}x^2\end{align*} and \begin{align*}y^2\end{align*} are both positive: not a hyperbola.
Both squared term coefficients are the same: a circle.
\begin{align*}r^2 = 9 \to r = 3\end{align*}
The center \begin{align*}(h, k)\end{align*} is \begin{align*}(1, -1)\end{align*}

2. To identify \begin{align*}3x^2 - 5y^2 + 23 = 0:\end{align*}

The x and y are both squared: not a parabola.
The \begin{align*}x^2\end{align*} and \begin{align*}y^2\end{align*} coefficients have different signs: a hyperbola.
\begin{align*}3x^2 - 5y^2 = -23 \to \frac{x^2}{5} - \frac{y^2}{3} = \frac{-23}{15}\end{align*}
Since the \begin{align*}y^2\end{align*} term is negative, it has a vertical transverse
The hyperbola's asymptotes are \begin{align*}y = \pm\frac{\sqrt3x}{\sqrt5}\end{align*}

3. To identify \begin{align*}-8x^2 - 18y^2 + 48x - 360y - 1800 = 0:\end{align*}

\begin{align*}-8(x - 3)^2 - 18(y + 10)^2 = -72\end{align*} ..... Complete the square
\begin{align*}\frac{(x - 3)^2}{9} + \frac{(y + 10)^2}{4} = 1\end{align*} ..... Simplify
Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are squared: not a parabola
Both coefficients are negative: not a hyperbola
The \begin{align*}x^2\end{align*} and \begin{align*}y^2\end{align*} terms do not have the same coefficients: not a circle either
This ellipse has center: (3, -10)
From the center, plot 2 up and \begin{align*}\sqrt9\end{align*} left and right to mark the vertices

4. To identify \begin{align*}2x^2 - 16x + 32 +3y = 6:\end{align*}

Only the \begin{align*}x\end{align*} term is squared, this is a parabola
\begin{align*}2(x-4)^2 + 3(y - 2)= 0\end{align*} ..... Factor and simplify
The vertex \begin{align*}(h, k)\end{align*} is \begin{align*}(4, 2)\end{align*}
The \begin{align*}x\end{align*} term is squared, this is an up/down parabola
The \begin{align*}x^2\end{align*} term is positive, this parabola opens down
The parent \begin{align*}y = x^2\end{align*} has been compressed vertically by 3 and horizontally by 2, resulting in a slightly wider than standard curve

### Practice

Identify the conic that is represented by the equation.

1. \begin{align*} \frac{(x - 5)^2}{4} + \frac{(y - 4)^2}{4} = 1\end{align*}
2. \begin{align*} (x - 3)^2 + y - 2^2 = 1\end{align*}
3. \begin{align*} x^2 - 5x - y^2 - 4y + 16 = 0\end{align*}
4. \begin{align*} \frac{2y}{3} = 2(x-4)^2 + 1\end{align*}

Convert to standard form:

1. \begin{align*}-8x^2 - 18y^2 + 48x - 360y - 1800 = 0\end{align*}
2. \begin{align*}-12x^2 + 12y^2 - 72x + 192y + 708 = 0\end{align*}
3. \begin{align*}x^2 + y^2 -8x + 10y + 32 = 0\end{align*}
4. \begin{align*}9x^2 + 4y^2 - 36x + 64y + 256 = 0\end{align*}
5. \begin{align*}9x^2 + y^2 + 90x - 8y + 232 = 0\end{align*}
6. \begin{align*}9x^2 - 9y^2 + 162x + 162y - 81 = 0\end{align*}

Identify and Graph the following:

1. \begin{align*}\frac{(x + 4)^2}{9} + (y)^2 = 1\end{align*}
2. \begin{align*}y = -2(x + 4)^2 - 4\end{align*}
3. \begin{align*}(x + 2)^2 - (y - 2)^2 = 1\end{align*}
4. \begin{align*}\frac{(x - 1)^2}{4} = \frac {(y)^2}{16} = 1 \end{align*}
5. \begin{align*}x = 2(y + 2)^2 - 1\end{align*}
6. \begin{align*}x = -(y - 2)^2 + 4\end{align*}
7. \begin{align*}\frac{(x + 2)^2}{1} - \frac{(y - 2)^2}{1} = 1\end{align*}
8. \begin{align*}\frac{(y - 1)^2}{4} - \frac{(x - 4)^2}{4} = 1 \end{align*}

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### Vocabulary Language: English Spanish

TermDefinition
tangent A line that intersects a circle in exactly one point.
Completing the Square Completing the square is a common method for rewriting quadratics. It refers to making a perfect square trinomial by adding the square of 1/2 of the coefficient of the $x$ term.
Conic Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
Ellipse Ellipses are conic sections that look like elongated circles. An ellipse represents all locations in two dimensions that are the same distance from two specified points called foci.
Ellipses Ellipses are conic sections that look like elongated circles. An ellipse represents all locations in two dimensions that are the same distance from two specified points called foci.
hyperbola A hyperbola is a conic section formed when the cutting plane intersects both sides of the cone, resulting in two infinite “U”-shaped curves.
Parabola A parabola is the characteristic shape of a quadratic function graph, resembling a "U".
Tangent line A tangent line is a line that "just touches" a curve at a single point and no others.

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