# 6.9: General Forms of Conic Sections

**At Grade**Created by: CK-12

**Practice**General Forms of Conic Sections

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.

Too bad Kelli didn't know about the material in this lesson!

### Watch This

Embedded Video:

- Khan Academy: Identifying Conics 1

### Guidance

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

Ellipses:

Parabolas:

Hyperbolas:

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:

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

The third equation is a parabola since there is an

But what about the fourth equation? Like the others, it is a degree-2 polynomial, since the exponents of the *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?

When

There are a few new, more general, rules I will show you that give more information about the case when

If

If

If

*Rotating Conics*

Suppose we have an equation of degree two polynomials, such as the

#### 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 *c** ^{2}* >

*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.

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

Convert to standard form:

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

Identify and Graph the following:

- \begin{align*}\frac{(x + 4)^2}{9} + (y)^2 = 1\end{align*}
- \begin{align*}y = -2(x + 4)^2 - 4\end{align*}
- \begin{align*}(x + 2)^2 - (y - 2)^2 = 1\end{align*}
- \begin{align*}\frac{(x - 1)^2}{4} = \frac {(y)^2}{16} = 1 \end{align*}
- \begin{align*}x = 2(y + 2)^2 - 1\end{align*}
- \begin{align*}x = -(y - 2)^2 + 4\end{align*}
- \begin{align*}\frac{(x + 2)^2}{1} - \frac{(y - 2)^2}{1} = 1\end{align*}
- \begin{align*}\frac{(y - 1)^2}{4} - \frac{(x - 4)^2}{4} = 1 \end{align*}

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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 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.### Image Attributions

Here you will explore the differences between the various equations of the conic sections, and will learn to identify a conic section by its equation.