This section covers:

**Tables of Conics****Circles****Applications of Circles****Parabolas****Applications of Parabolas****Ellipses****Applications of Ellipses****Hyperbolas****Applications of Hyperbolas****Identifying the Conic****More Practice**

**Conics** (**circles**, **ellipses**, **parabolas**, and **hyperbolas**) involves a set of curvesΒ that are formed by intersecting a plane and a double right cone (probably too much information!).Β But in case you are interested, there are four curves that can be formed, and all are used in applications of math and science:

In the Conics section, we will talk about each type of curve, how to recognize and graph them, and then go over some common applications (sorry β another way of sayingΒ βword problemsβ).

**Always draw pictures first when working with Conics problems!**

# Table of Conics

Before we go into depth with each conic, here are the **Conic Section Equations**. Β Note that you may want to go through the rest of this section before coming back to this table, since it may be a little overwhelming at this point!

Note that we can also write equations for circles, ellipses, and hyperbolas in terms of **cos** and **sin**, and other **trigonometric**Β functions; there are examples of these in the **Introduction to Parametric Equations** section.

# Circles

Youβve probably studied **Circles** in **Geometry** class, or even earlier.

Circles are defined as a set of points that are equidistant (the same distance) from a certain point; this distance is called the **radius** of a circle.

Here is the equation for a circle, where** r** is the radius: Β Β .

If we were to solve for ** y** in terms of

**(for example, to put in the graphing calculator), weβd get: Β Β Β .**

*x*Here are graphs of sample circles, with their domains and ranges:

Sometimes we have to **complete the square** to get the equation for a circle.Β We learned how to complete the square with quadratics here in the **Factoring and Completing the Square** Section.

**Problem:**

Find the **center** and **radius** of the following **circle**: Β Β \({{x}^{2}}+{{y}^{2}}-6x-12y-55=0\).

**Solution:**

You may see a βtrickβ problem like this, where the solution is just a point:

**Problem:**

**Complete the square** and **graph**: Β Β \({{x}^{2}}+{{y}^{2}}-4x+2y+5=0\).

**Solution:**

## Writing Equations ofΒ Circles

Sometimes you will have to come up with the equations of circle, or tangents of circles.

**Problem:**

Write the **equation** **of the line** that is **tangent** to the **circle**Β \({{\left( x-3 \right)}^{2}}+{{\left( y+2 \right)}^{2}}=61\) Β at the pointΒ \(\left( -2,\,\,-8 \right)\).

**Solution:**

A line **tangent** to a circle means that it touches the circle at one point on the outside of the circle, at a radius that is perpendicular to that line:

For this problem, since we only have one point on the tangent line (β2, β8), Β we’ll have to get the **slope of the line** of the line to get its equation.Β Remember here in the **Coordinate System and Graphing Lines**Β that perpendicular lines have slopes that are **opposite reciprocals** of each other.

Letβs **draw a picture**, and then get the solution:

Hereβs another type of problem you might see:

**Problem: **

The lines Β \(y=\frac{4}{3}x-\frac{5}{3}\) Β andΒ Β \(y=-\frac{4}{3}x-\frac{{13}}{3}\) Β each contain diameters of a circle, and the point Β \(\left( {-5,\,\,0} \right)\) Β is also on that circle.Β **Find the equation of this circle**.

**Solution:**

If two lines are **both diameters** of the same circle, where they intersect must be the **center** of the circle.Β In this case, it was easier to **draw a picture** to see that this is true:

So now we can get the center of the circle by finding the intersection of the two lines.Β Since we have another point, too, we can get the equation of the circle:

## Applications of Circles

**Problem:**

A pizza delivery area can be represented by a circle, and extends to the points **(0, 18)** and **(β6, 8)** (these points are on the **diameter** of this circle).Β Write an equation for the circle that models this delivery area.Β

**Solution:**

If we draw a picture, weβll see that weβll have to use both the **Distance Formula** and **Midpoint Formula**Β from the **Coordinate System and Graphing Lines** section.

Letβs **draw a picture**, and then get the solution:

# Parabolas

Letβs revisit **parabolas** (a type of quadratic), but go into a little more depth here.

We studied Parabolas in the **Introduction to Quadratics** section, but we only looked at βverticalβ parabolas (that either go up or down).

We remember that a parabola is in the form Β \(y=a{{\left( x-h \right)}^{2}}+k\), where (*h*, *k*) is the **vertexΒ **and β** x = h**β is the

**axis of symmetry**or

**Β line of symmetry (LOS)**; this is a βverticalβ parabola. Β Β Note that this can also be written Β \(y-k=a{{\left( {x-h} \right)}^{2}}\) Β or Β \(b\left( {y-k} \right)={{\left( {x-h} \right)}^{2}}+k\), where Β \(b=\frac{1}{a}\).

ParabolasΒ can also be in the form Β \(x=a{{\left( y-k \right)}^{2}}+h\), where (*h*, *k*) is the **vertex**, and β** y = k**β Β is the

**LOS**; this is a “horizontal” parabola

**.Β**The

**lineΒ of symmetry (LOS)**is a line that divides the parabola into two parts that are mirror images of each other.

Technically, a parabola is the set of points that are equidistant from a line (called the **directrix**) and another point not on that line (called the **focus, **or** focal point**).

So ifΒ * p*Β is the distance from theΒ

**vertex**Β to theΒ

**focus**point

**Β**(called the

**Β focal length**), it isΒ also the distance from theΒ

**vertex**Β to theΒ

**directrix. Β**This makesΒ the distance from theΒ

**focus**Β to theΒ

**directrix**Β isΒ

**2**. Β Note that the focus is always βinsideβ the parabola on the line of symmetry, and the directrix is βoutsideβ the parabola.

*p*Also note that the line perpendicular to the line of symmetry (and thus parallel to the directrix) that connects the **focus** **to the sides of the parabola** is called the **latus chord**, **latus rectum**, or**Β focal width, focal diameter, focal chord **or** focal rectum**; the length of this chord is **4 p**.Β You can use this information to help you draw the parabola, if you know the distance from the vertex to the focus (

**).**

*p*Here is a parabola with **center (0, 0)**:

IIf the vertex is at the origin (0, 0), the equation of the parabola is Β \(y=a{{x}^{2}}\), and Β \(a=\frac{1}{4p}\); Β if you do the algebra, it follows that that Β \(p=\frac{1}{4a}\).

For example, if ** p = 4Β **(length of focus to vertex), Β the equation of the parabola would be Β \(y=\frac{1}{4\left( 4 \right)}{{x}^{2}}\,\,=\,\,y=\frac{1}{16}{{x}^{2}}\).

Note that the above graph shows a parabola that has a positive value before the Β \({{x}^{2}}\).

Here are the four different βdirectionsβ of parabolas and the generalized equations for each:

Note that sometimes (as in the problem below) we have to **complete the square**Β to get the equation in parabolic form; we did this here in the **Solving Quadratics by Factoring andΒ Completing the Square** Section.

Letβs do some problems!

**Problem:**

Identify the **vertex**, **axis of symmetry**, **focus**, **equation of the directrix, **and** domain **and** range** for the following parabolas, then **graph the parabola**:Β (a)Β \(y-4=2{{\left( x-3 \right)}^{2}}\) Β (b)Β \(x=-\frac{{{y}^{2}}}{2}+2y+4\).

**Solution:**

Itβs typically easier to **graph the parabola** first, and then answer the questions.

## Writing Equations of Parabolas

**Problem:**

Write the **equation of a parabola** with a **vertex** of** **( β2, 4)** **and **focus poin**t (0, 4). Β Β Also find the **domain** and **range** of the parabola.

**Solution:**

## Applications ofΒ Parabolas

**Problem:**

The equation Β \(\frac{1}{32}{{x}^{2}}\) Β models cross sections of parabolic mirrors that are used for solar energy.Β There is a heating tube located at the focus of each parabola; **how high is this tube located above the vertex of its parabola**?

**Solution:**

For problems like these, unless otherwise noted, just assume the vertex of the parabola is at **(0, 0)**.Β Since we know that the equation of a parabola is Β \(y=a{{x}^{2}}\), where Β \(a=\frac{1}{4p}\)Β and Β \(p=\frac{1}{4a}\), then for Β \(\frac{1}{32}{{x}^{2}}\), we have Β \(\frac{1}{32}=\frac{1}{4p}\).Β We can either cross multiply or just do mental math to see that ** p = 8**.Β So the heating tube needs to be located

**8 units**above the vertex of the parabola.

**Problem:**

A searchlight has a parabolic reflector (has a cross section that forms a βbowlβ).Β The parabolic βbowlβ is 16 inches wide from rim to rim and 12 inches deep.Β The filament of the light bulb is located at the focus. Β (a)Β **What is the equation of the parabola used for the reflector?**Β Β Β (b)Β **How far from the vertex is the filament of the light bulb?**

**Solution:**

Letβs graph this particular parabola, again putting the vertex at **(0, 0)**.

**Problem:**

The cables of the middle part of a suspension bridge are in the form of a parabola, and the towers supporting the cable are 600 feet apart and 100 feet high. Β **What is the height of the cable at a point 150 feet from the center of the bridge?**Β

**Solution:**

Letβs draw a picture of the bridge, and place the middle of the cable (vertex) at the point **(0, 0)**.

# Ellipses

An **ellipse** sort of looks like an oval or a football, and is the set of points whose distances from two fixed points (called the foci) inside the ellipse is always the same, Β \({{d}_{1}}+{{d}_{2}}=2a\). Β The distance **2 a** is called the

**constant sum**or

**focal constant**, andΒ

**is the distance between the center of the ellipse to a**

*a***vertex**(you usually donβt have to worry about the Β \({{d}_{1}}\) Β and Β \({{d}_{2}}\)).Β

**Β (the length of the center to the**

*a***vertices**) is always bigger than

**(the length of the center to the**

*b***co-vertices**).

The equation of a βhorizontalβ ellipse that is centered on the origin **(0, 0)** is Β \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\) Β (what’s under the * x* is larger than what’s under the

*). Β The length of the longest axis (called the*

**y****major axis**) is alwaysΒ

**2**, and this is along the

*a***for a horizontal ellipse. Β Again, the distance from the center of the ellipse to a vertex is**

*x*-axis**Β so the vertices are at Β \(\left( \pm a,\,\,0 \right)\).**

*a,*The length of the smaller axis (called the **minor axis** or) is **2 b**, and this is along the

**for a horizontal ellipse. Β Again, the distance from the center of the ellipse to a co-vertex is**

*y*-axis**Β so the co-vertices are at Β \(\left( 0,\,\,\pm b \right)\).**

*b,Β*The focuses or **foci** always lie inside the ellipse on the **major** **axis**, and the distance from the center to a focus is ** c**.Β So the foci are at Β \(\left( \pm c,\,\,0 \right)\) Β for this type of ellipse, and it turns out that Β \({{a}^{2}}-{{b}^{2}}={{c}^{2}}\).

Note that a circle happens when ** a** and

**are the same in an ellipse, so a circle is a special type of ellipse, but for all practical purposes, circles are different than ellipses.Β Sometimes you will be asked to get the**

*b***eccentricity**of an ellipse Β \(\frac{c}{a}\), which is a measure of how close to a circle the ellipse is; when it is a circle, the eccentricity is 0. Β Also, the

**area**of an ellipse is Β \(\pi ab\).

Note also that the **focal width** (**focal chord**, or **focal rectum**) of an ellipse is Β \(\frac{{2{{b}^{2}}}}{a}\);Β this the distance perpendicular to the major axis that goes through the focus.

Here is a **horizontal ellipse**; we will also look at vertical and transformed ellipses below.

Here are the two different βdirectionsβ of ellipses and the generalized equations for each:

You also may have to **complete the square** to be able to graph an ellipse, like we did **here** for a circle. Β (And since you always have to have a β**1**β after the equal sign, you may have to divide all terms by the constant on the right, if it isnβt β**1**β).

Letβs put it all together and graph some ellipses:

**Problem:**

Identify the **vertices**, **co-vertices**, foci, and **domain** and **range**Β for the following **ellipses**; then graph: (a) Β \(9{{x}^{2}}+49{{y}^{2}}=441\)Β Β Β (b) Β Β \(\frac{{{\left( x+3 \right)}^{2}}}{4}+\frac{{{\left( y-2 \right)}^{2}}}{36}=1\).

**Solution:**

Itβs typically easier to **graph the ellipse first**, and then answer the questions:

Hereβs one where you have to **Complete the Square** to be able to graph the ellipse:

**Problem:**

Identify the **vertices**, **co-vertices**, **foci** andΒ **domain** and **range**Β for the following **ellipse**; then **graph**:Β Β \(4{{x}^{2}}+{{y}^{2}}+24x+2y=-33\).

**Solution:**

## Writing Equations of Ellipses

You may be asked to **write an** **equation **from either a graph or a description of an ellipse:

**Problem**

Write the **equation of the ellipse**:

**Solution:**

We can see that the ellipse is 10 across (the **major axisΒ length**) and 4 down (the **minor axis length**).Β So, 2*a* = 10, and 2*b* = 4. Β Β We can also see that the center of the ellipse (*h*, *k*) Β is at (4, β3).

Since the ellipse is **horizontal**, weβll use the equation Β \(\frac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}+\frac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1\).Β Plug in our values for ** a**,

**,**

*b***, and**

*h***, and we get Β \(\frac{{{\left( x-4 \right)}^{2}}}{25}+\frac{{{\left( y+3 \right)}^{2}}}{4}=1\).Β Not too bad!**

*k***Problem:**

Find the **equation of this ellipse**, **graph**, and find theΒ **domain** and **range**:Β Endpoints of major or minor axis at (β1, β6) and (β1, 2) and focus at (β1, β3).Β

**Solution:**

Letβs **graph the points we have**, and go from there.

## Applications ofΒ Ellipses

The **foci of ellipses** are very useful in science for their **reflective properties** (sound waves, light rays and shockwaves, as examples), and are even used in medical applications.Β Β Β In fact, Keplerβs first law of planetary motion states that the **path of a planetβs orbit models an ellipse** with the sun at one focus, so the orbits of asteroids and other bodies are another elliptical application.

**Problem:**

Two girls are standing in a whispering gallery that is shaped like semi-elliptical arch.Β The height of the arch is 30 feet, and the width is 100 feet.Β Β **How far from the center of the room should whispering dishes be placed so that the girls can whisper to each other? Β **(Whispering dishes are places at theΒ **foci**Β of an ellipse).

**Solution:**

**Problem:**

An ice rink is in the shape of an ellipse, and is 150 feet long and 75 feet wide.Β **What is the width of the rink 15 feet from a vertex?**

## Hyperbolas

A **hyperbola **sort of looks like two parabolas that point at each other, and is the set of points whose distances from two fixed points (the foci) inside the hyperbolaΒ is always the same, Β \({{d}_{1}}-\,\,{{d}_{2}}=2a\). Β The distance **2 a** is called the

**focal radii**

**Β distance**,

**focal constant,Β**or

**constant difference**, and Β

**is the distance between the center of the hyperbola to a**

*a***vertex**.

The equation of a βhorizontalβ hyperbola (as shown below) that is centered on the origin **(0, 0)** is Β \(\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\). Β The length of the axis in which the hyperbola lies (called the** transverse axis**) is **2 a**, and this is along the

**for a horizontal hyperbola. Β Again, the distance from the center of the hyperbolaΒ to a vertex isΒ**

*x*-axis**,**

*a*

*Β**s*o the vertices are at Β \(\left( \pm a,\,0 \right)\).

The length of the **conjugate axis** is **2 b**, and note that

**does**

*a***not have to be bigger**than

**, like it does for an ellipse.Β (The distance from the center of the hyperbolaΒ to a co-vertex isΒ**

*b**). Β Also note where the*

**b****is not on the hyperbola; it is on what we call the**

*b***central rectangle**Β (or

**fundamental rectangle**) of the hyperbola (whose

**diagonals are asymptotes**for the hyperbola).Β Β So the

**conjugate axis**is along the

**for a horizontal hyperbola, and the co-vertices are at Β \(\left( 0,\,\pm b \right)\).**

*y*-axisThe **asymptotes** for a horizontal hyperbola centered at (0, 0) are Β \(y=\pm \frac{b}{a}x\)Β ( \(\pm \frac{b}{a}\) Β are the **slopes**, orΒ the square root of whatβs under the ** y** over the square root of whatβs under the

**). Β The asymptotes are the**

*x***diagonals**of the central rectangle of the hyperbola.

The focuses or **foci** always lie inside the curves on the **major** **axis**, and the distance from the center to a focus is ** c**.Β So the foci are at Β \(\left( \pm c,\,\,0 \right)\) Β for a horizontal hyperbola (like an ellipse!), and it turns out that Β \({{a}^{2}}+{{b}^{2}}={{c}^{2}}\).Β Β Β (I like to remember that you always use the different sign for this equation: since ellipses have a

**plus sign**in the equation Β \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\), they have a

**minus sign**in Β \({{a}^{2}}-{{b}^{2}}={{c}^{2}}\) ; since hyperbolas have a

**minus sign**in the equation Β \(\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\), they have a

**plus sign**in Β \({{a}^{2}}+{{b}^{2}}={{c}^{2}}\).)

Sometimes you will be asked to get theΒ **eccentricity**Β of an hyperbola Β \(\frac{c}{a}\), which is a measure of how βstraightβ or βstretchedβ the hyperbola is.

Note also that, like for an ellipse, the **focal width** (**focal chord**, or **focal rectum**) of an ellipse is Β \(\frac{{2{{b}^{2}}}}{a}\);Β this the distance perpendicular to the major axis that goes through the focus.

Here is a** horizontal hyperbola**; we will also look at verticalΒ and transformed hyperbolas below.

Here are the two different βdirectionsβ of hyperbolas and the generalized equations for each:

You also may have toΒ **complete the square**Β to be able to graph an hyperbola, like we didΒ **here**Β for a circle. Β (And since you always have to have a β**1**β after the equal sign, you may have to divide all terms by the constant on the right, if it isnβt β**1**β).Β Remember, for the conic to be a hyperbola, the coefficients of the Β \({{x}^{2}}\) Β and Β \({{y}^{2}}\) Β **must have different signs**.

Letβs put it all together and graph some hyperbolas:

**Problem:**

Identify theΒ **center**, **vertices**, **foci**, andΒ **equations of the asymptotes** for the following **hyperbolas**; then **graph**: (a) Β \(9{{x}^{2}}-16{{y}^{2}}-144=0\)Β Β (b) Β \(\frac{{{\left( y+3 \right)}^{2}}}{4}-\frac{{{\left( x-2 \right)}^{2}}}{36}=1\).

**Solution:**

Itβs typically easier to **graph the hyperbola first**, and then answer the questions.

Hereβs one where you have to **Complete the Square** to be able to graph the hyperbola:

**Problem**:

Identify the theΒ **center**, **vertices**, **foci**, andΒ **equations of the asymptotes** for the following **hyperbola**; then **graph**: Β \(49{{y}^{2}}-25{{x}^{2}}+98y-100x+1174=0\).

**Solution:**

## Writing Equations of Hyperbolas

You may be asked **to write an** **equation **from either a graph or a description of a hyperbola:

**Problem:**

Write the **equation of the hyperbola**:

**Solution:**

We can see that the center of the hyperbola is (2, β5), the **transverse axis lengthΒ **(** 2a**)

**Β**is 6, and the

**conjugate axis length**Β (

**2**

*) is also 6.Β Β So*

**b****= 3, and**

*a***= 3.**

*b*Since the hyperbola is **horizontal**, weβll use the equation Β \(\frac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}-\frac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1\). Β Plug in our values, and we get Β \(\frac{{{\left( x-2 \right)}^{2}}}{9}-\frac{{{\left( y+5 \right)}^{2}}}{9}=1\).

**Problem:**

Find the equation of the **hyperbola** where the **difference of the focal radii** isΒ 6, and the **endpoints of the conjugate axis** are (β2, 8) and (β2, β2).

**Solution:**

We probably donβt even need to graph this hyperbola, since weβre basically given what ** a** and

**are.Β Β Remember that the**

*b***difference of the focal radii**is

**2**, so

*a***= 3.**

*a*Since the endpoints of the conjugate axis are along a vertical line, we know that the hyperbola is **horizontal**, and the **co-vertices** are (β2, 8) and (β2, β2).Β From this information, we can get the center (midpoint between the co-vertices), which is (β2, 3) and the length of the minor axis (2*b*), which is 10.Β So *b* = 5.Β Β (Draw the points first if itβs difficult to see).

So the equation of the ellipse is Β \(\frac{{{\left( x+2 \right)}^{2}}}{9}-\frac{{{\left( y-3 \right)}^{2}}}{25}=\,\,1\).

**Problem:**

Find the equation of the **hyperbola** where one of the **vertices **is at Β \(\left( {-3,2} \right)\), and the **asymptotes **are **Β **\(y-2=\pm \frac{2}{3}\left( {x-3} \right)\).

**Solution:**

Letβs try to graph this one, since itβs hard to tell what we know about it!

We can see from the equation of the asymptotes that the **center** of the hyperbola is Β \(\left( {3,2} \right)\).

Then weβll graph this **center** and also graph the **vertex** that is given to see that the hyperbola is **horizontal**:

## Applications of Hyperbolas

Like ellipses, the **foci of hyperbolas** are very useful in science for their **reflective properties**, and hyperbolic properties are often used in telescopes.Β Β They are also used to **model paths of moving objects**, such as alpha particles passing the nuclei of atoms, or a spacecraft moving past the moon to the planet Venus.

**Problem:**

A cometβs path (as it approaches the sun) can be modeled by one branch of the hyperbola Β \(\frac{{{y}^{2}}}{1096}-\frac{{{x}^{2}}}{41334}=\,\,1\), Β where the sun is at the focus of that part of the hyperbola. Β Each unit of the coordinate system is 1 million miles. (a)Β Β Find the coordinates of the sun (assuming it is at the focus with non-negative coordinates).Β Round to the nearest hundredth. Β Β (b)Β Β How close does the comet come to the sun?

**Solution:**

Again, itβs typically easier to **graph the hyperbola first**, and then answer the questions.

**Problem:**

Two buildings in a shopping complex are shaped like a branches of the hyperbola Β \(729{{x}^{2}}-1024{{y}^{2}}-746496=0\)Β , where *x* and *y* are in feet.Β How far apart are the buildings at their closest part?

**Solution:**

Letβs try this one without drawing it, since we know that the closest points of a hyperbola are where the vertices are, and the buildings would be **2 a** feet apart.

By doing a little algebra (adding 746496 to both sides and then dividing all terms by 746496) we see that the equation in hyperbolic form is Β \(\frac{{{x}^{2}}}{1024}-\frac{{{y}^{2}}}{729}=1\).Β Β So ** a**Β =Β \(\sqrt{1024}\) = 32.Β So the building are 32 x 2 =

**64 feet apart**at their closest part.

**Problem:**

Two radar sites are tracking an airplane that is flying on a hyperbolic path.Β The first radar site is located at (0, 0), and shows the airplane to be 200 meters away at a certain time.Β The second radar site, located 160 miles east of the first, shows the airplane to be 100 meters away at this same time.Β Find the coordinates of all possible points where the airplane could be located.Β (Find the **equation of the hyperbola** where the plane could be located).

**Solution:**

Let’s draw a picture first andΒ remember that the **constant difference** for a hyperbola is always **2 a. Β Β **The planeβs path is actually on one branch of the hyperbola; letβs create a horizontal hyperbola, so weβll use the equation Β \(\frac{{{\left( x-h \right)}^{2}}}{{{a}^{2}}}-\frac{{{\left( y-k \right)}^{2}}}{{{b}^{2}}}=1\):

**Problem:**

Alpha particles are deflected along hyperbolic paths when they are directed towards the nuclei of gold atoms.Β If an alpha particle gets as close as 10Β units to the nucleus along a hyperbolic path with asymptote Β \(y=\frac{2}{5}x\), what is the equation of its path?

**Solution:**

Letβs draw a picture first andΒ make the nucleus the center of the hyperbola at (0, 0).

# Identifying the Conic

Sometimes you are given an equation or a description of a conic, and asked to identify the conic.Β Β Remember these rules:

- Β \({{x}^{2}}\) Β with other
*y*βsΒ (and maybe*x*βs), Β or Β \({{y}^{2}}\) Β with otherΒ*x*βsΒ (and maybe*y*βs): Β Β**parabola** - Β \({{x}^{2}}\) Β and Β \({{y}^{2}}\)Β with same coefficients and Β + Β sign:Β
**Β Β circle** - Β \({{x}^{2}}\) Β and Β \({{y}^{2}}\)Β with same coefficients and Β β Β sign:Β Β
**Β hyperbola** - Β \({{x}^{2}}\) Β and Β \({{y}^{2}}\) Β with different coefficients Β and Β + Β sign:Β Β
**Β ellipse** - Β \({{x}^{2}}\) Β and Β \({{y}^{2}}\) Β with different coefficients and Β β Β sign:Β Β
**Β hyperbola**

Here are some examples; I always find itβs easier to work/graph these on graph paper to see whatβs going on:

For the following, **write the equation of the conic, using the given information**:

**Learn these rules, and practice, practice, practice!**

Hit Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view stepsβ, you will go to theΒ **Mathway**Β site, where you can register for theΒ **full version**Β (steps included) of the software. Β You can even get math worksheets.

You can also go to theΒ **Mathway**Β site here,Β where you can register, or just use the software for free without the detailed solutions. Β There is even a Mathway App for your mobile device. Β Enjoy!

On to **Systems of Non-Linear Equations****Β Β **β you are ready!

This is simply amazing and very helpful.Lisa,can u please complete these to be completed sections as well.I love math too.

Thank you so much.

Hi Lisa, excellent website and very clear lessons!

My younger sister has to prepare for a college entry exam in our country but she hasn’t studied Conics before. (Neither have I, so I’m searching the net about this topic to learn about it myself!) She has a couple of weeks to the exam. I was wondering – would you by any chance be able to complete this page before then?

Thank you so much!

Thanks so much for writing! I’m hoping to have the Conics section finished this week π

Lisa

I teach conic sections in my Pre-Calculus class. This website will be a great tool for students who can work ahead of me (they can learn independently by reading your information) or for those who need remediation and review. I plan to try using it with such students this year. Thanks for the great information. Nice job.

Renee,

Thank you SO much for writing; notes like yours make me want to write more and finish! Please let me know if you see any errors and see ways to make the site better. Happy New Year! Lisa

This is a GREAT Conics Reference tool with the examples! I really like the “Table of Conics” visual at the beginning of the page. Just a quick note I think the Horizontal and Vertical Parabola Equations on the “Table of Conics” might be switched. Once again great information and thanks for organizing it and making it more understandable.

Janel,

THANKS so much for finding this error – I really appreciate it! Let me know if you see anything else, OK? And I’m glad you like the site – please spread the word about She Loves Math π Lisa

There is an error on the sideways parabola notes. The axis of symmetry is y = k, not h. The axis of symmetry is y = k, not h. I like the word problems.

Thanks so much for finding this! Please let me know if you see anything else. I’ve corrected it; I think the problem was in the formula actually. Thanks again π Lisa

The problem with the airplane and the hyperbola is interesting, but it’s misleading. If the airplane is 200 miles from one radar site and 100 from the other, there are only TWO possible positions for the airplane (the vertex of a triangle with lengths of 160, 100, and 200). The wording could be changed to reflect that the airplane is known to be following a hyperbolic path (maybe it’s a rocket being launched from one of the vertices and the distances are being measured as distance in the x and altitude in the y, in meters instead?), or else to say that, instead of the airplane being 200 and 100 miles away, there is a constant difference in radar signal reception between the two sites and the airplane?

Brian,

Thanks so much for writing – great point! I’ll rework the problem to fix the mistake π Please let me know if you see anything else! Lisa

Hello I found this website EXTREMELY HELPFUL! The explanations were clear, concise and I really appreciate the tables that show each conic section horizontal/vertical with positive and negative coefficients. This must have taken a great deal of work, which I think it is very well done. Great job!!

Thank you so much for taking the time out to write!! This makes me want to write faster and finish the site. Thanks again, and let me know if you have any suggestions for the site! Lisa

thanks for keeping this videos and images

Hello, and thank you for sharing this write up. I’m a third level Calculus student and this is the PERFECT refresher for what I’m dealing with in class. I am eternally grateful!

Thank you so much for this website! I’ve used it a lot while studying for my final exam. I really appreciate all the example problems and the well-explained steps and tips. Conics were always a really hard unit for me and your website has cleared it up quite a bit.

Thanks so much for writing, and thanks for using She Loves Math! You made my day! Comments like yours make me want to keep writing π Lisa

Great lessons. I have specifically liked the conics lessons in here! Very clearly explained with excellent examples. Thank you !

Thanks so much! Please spread the word about She Loves Math! Lisa π

Thank you very much it provided me help when I lost my notes and tomorrow is my paper so thank you very much.

What is the application or importance of four conic section in aeronautical engineering course?

Thanks for writing! I’m not an aeronautical engineer, but I saw in a textbook that they can be used to model the orbits of satellites when they escape the earth’s gravity. I can’t tell you specifics though – sorry! π Lisa

same error under math note twice in section regarding asymptotes, it states “remember to use the square root of whats under the y for the numerator of the of the slope, and the square of whats under the x for the denominator. Should be … the square root of whats under the x for the denominator

Hello, and thanks for writing! Can you double-check on this? I thought it’s like a slope – where y is on the top and x is on the bottom. Thanks, Lisa

I just want to give a lot of thanks to you, Lisa! This page is just absolutely wonderful. I learned everything I need. This one’s the best place I’d ever been, and I hope that I’ll be coming back here soon. Good luck to me for our midterm examination π such a big help, sissy.

Thanks so much! I’m sure you’ll do great on your midterm. π Lisa

Thank you very much. This is very well done, thorough, and consistent. The wide range of problems is a great selection of the type of problems out there.

I believe there are a couple of typos, may be you can take a look. The first example (a) for hyperbola in the solution it states the foci are (0,+- 5), but I believe it should be (+-5,0) as per your graph. Also the domain has (-infinity,4], but I believe you left out the minus sign, it should be -4.

Just to make this very well put together document, a little better.

Thanks again,

RK

Thanks you SO MUCH for finding these – let me know if you see anything else, please!!! π Lisa

Your welcome,

How about this, in the beginning of the hyperbola section it says ” inside the ellipse is always the same” is that supposed to be hyperbola?

Thanks so much again. I’ve obviously been looking at this stuff too long π Lisa

WRONG EQUATION!!!!

x= up down (Vertical)

y= left and right (Horizontal)

because of this i FAILED my EXAMINATION!!!

Oh gosh, I’m SO sorry!!! I may have a typo! Can you tell me where the wrong equation is?? Lisa

hi it a nice

send me some pdfs

PDFs of what? I’m thinking about putting a link to PDFs on my website – do you think there’d be interest? Lisa

woww i hv really liked it and as well understood.

can you please help me out with this 36x^2 + 24xy + 29y^2 + 126y – 72x + 81 =0

Thanks for writing! I think this problem is a little beyond the scope of this website, but I think there’s a solution here:

https://books.google.com/books?id=0VLMn_OHpcgC&pg=PA125&lpg=PA125&dq=36x%5E2+%2B+24xy+%2B+29y%5E2+%2B+126y+%E2%80%93&source=bl&ots=4-xi3NdHQt&sig=y9FlxakuSaRfH_UiDAcdZPYy2Iw&hl=en&sa=X&ved=0ahUKEwi52JXnlJrQAhVHlFQKHdiDDdgQ6AEIHjAA#v=onepage&q=36x%5E2%20%2B%2024xy%20%2B%2029y%5E2%20%2B%20126y%20%E2%80%93&f=false

Lisa

A wonderfull

Maths questions

Thank you!

hey i just saw your website and i am impressed with what you are doing keep up and i wish you the best. And again happy new year

Thanks so much for your note; that means a lot to me. Please spread the word about shelovesmath.com π Happy New Year’s! Lisa

Please can you help me with this question?

The equation of an ellipse E is given x^2/a^2 + y^2/b^2 = 1 where a and b are positive constants. The ellipse E intercept the x and y axis at the point A and B respectively. Determine the equation of the perpendicular bisector L of AB. Given that L cuts the x axis at P and that M is the midpoint of AB.

Proof that the area of triangle PMA is b(a^2 + b^2)/8a

Given that a^2 =3b^2,find in terms of a and b the coordinates of the point where the line L cuts the ellipse E

Thanks for writing! I got y = (a/b)x + (b^2-a^2)/(2b) for the first answer; I was able to prove the second, and got (b^2/a, 0) for the third. Did you get these? Let me know if you want to see the work. Thanks! Lisa