Picture a solid cone in your head. Or better yet, have a look at the one below. If you were able to take a big butcher knife and slice through the cone, what shape would be left by the cut? Well, you may be thinking, “Depends on where you cut.” And you’d be right. If you cut through the cone perfectly horizontally (assuming the cone is sitting flat on the table), the cut would look like a circle. If you angle the blade just slightly, the cut will now look more oval-shaped (in Geometry, this is called an ellipse). If you slice straight downward through the side, the cut would look like an arch, or a parabola. These cross-sections are the basis for conic sections.
Fig. 1 - Single cone.
Conic Sections Definition
The image above is of a simple cone, but when you talk about conic sections, you actually need to be thinking of a double cone, like in the image below. This becomes important as it introduces one more cross-section that you will learn about.
Fig. 2 - Double cone used for conic sections.
A conic section is the resulting curve when a cone intersects with a plane.
Conic sections can be thought of as families of curves that result from intersecting a plane with a particular cone.
Types of Conic Sections
There are four types of conic sections that can result from these intersections.
Circle
When the plane intersects the cone perpendicular to the axis (but not through the center point), the resulting cross-section will be a circle. Circles are technically a specific type of ellipse.
Fig. 3 - Circle formed by the intersection of a cone and a plane that is perpendicular to the axis of the cone.
Ellipse
When the plane intersects one of the cones at an incline, the resulting cross-section will be an ellipse. Ellipses (and therefore circles as well) are considered closed conic sections.
Fig. 4 - Ellipse formed by the intersection of a cone and an inclined plane.
For more information on this type of conic section, see Ellipses.
Parabola
When the plane intersects one of the cones through one of the bases, the resulting cross-section will be a parabola. A parabola is an unbounded conic section.
Fig. 5 - Parabola formed by the intersection of a cone and a plane that passes through one of the bases.
Hyperbola
When the plane intersects both cones (but does not pass through the center), the resulting cross-section is a hyperbola. A hyperbola is made of two pieces, called branches, that look like two symmetrical parabolas. Hyperbolas are also unbounded conic sections.
Fig. 6 - Hyperbola formed by the intersection of a double cone and a plane that passes through the bases of both cones.
Conic Sections Graph
Each conic section can be defined by an equation that can be graphed on a standard Cartesian coordinate plane. But before looking at the equations, let’s look at their graphs and some of the important features.
Features of all conic sections
All conic sections have three features in common: a focus (or foci), a directrix (or directrices), and an eccentricity.
Focus
A focus is a special, fixed point used in the construction of conic sections on a coordinate plane. It is located "inside" the conic section. You may also see the foci called loci.
Circles and parabolas have one focus. Ellipses and hyperbolas have two foci (the plural for the word focus). Together with the directrix, the focus helps determine the eccentricity and curvature of the conic section.
For more information on these topics, see Eccentricity of Conic Sections and Loci with Conic Sections.
Directrix
A directrix is a fixed line that is perpendicular to the axis of the conic section that, together with the foci, help define the shape of the conic section. It is located “outside” the conic section.
A parabola has one directrix. Ellipses and hyperbolas have two directrices (the plural for the word directrix). A circle does not have a defined directrix. You can think of the distance between the points on a circle and its “directrix” as being infinite.
Eccentricity
The eccentricity describes the curvature of the conic section and is defined by the ratio of the distance between a point on the conic section and a focus to the distance between that point and the directrix.
The eccentricity will be constant within a conic section. A higher eccentricity means a lower curvature, because the eccentricity tells you how much the conic section varies from being a circle. The size of the eccentricity can also tell you which type of conic section you are working with:
If the eccentricity is equal to \(0\), then the conic section is a circle.
If the eccentricity is between \(0\) and \(1\), then the conic section is an ellipse.
If the eccentricity is equal to \(1\), then the conic section is a parabola.
If the eccentricity is greater than \(1\), then the conic section is a hyperbola.
To further understand eccentricity and the reasoning between these values, see our article on the Eccentricity of Conic Sections.
Circle
As usual, circles are rather elegantly simple. You only need to know the center point \((h,k)\) and the radius \(r\). With that information, you can graph a circle and write its equation. For a circle, the center is also the focus, and, as mentioned above, there is no defined directrix. The image below shows an example of a circle with the center and the radius labeled.
Fig. 7 - Graph of a circle showing center and radius.
Ellipse
Ellipses have many features that are similar to circles. In fact, circles are a particular kind of ellipse. A typical ellipse looks like what you would think an oval looks like. They are wider in one direction than the other. The wider direction is called the major axis, and the shorter direction is called the minor axis.
To further explore ellipses’ features and equations, check out our article on Ellipses. The image below shows an ellipse with the center, foci, and directrices labeled.
Fig. 8 - Graph of ellipse showing center, foci, directrices, and major and minor axes.
Parabola
You may already be somewhat familiar with parabolas since they are a major topic in Algebra. You probably know that a parabola looks like a symmetrical arch or u-shape. In addition to a focus and a directrix, a parabola has a vertex. This is located on the parabola's axis of symmetry at the "turn".
Fig. 9 - Graph of a parabola showing vertex, focus, and directrix.
Hyperbola
A hyperbola looks like a matching pair of parabolas going outward in opposite directions. For hyperbolas, the center \((h,k)\) is located equidistant from the two vertices of the branches. The distance between the vertices of the two branches is called the transverse axis. The conjugate axis is perpendicular to this. In the next section, you will see how the equation affects these values. These axes also help determine the slant asymptotes that form the shape of the parabola.
Fig. 10 - Graph of hyperbola showing center, vertices, foci, directrices, and transverse and conjugate axes.
Solving Conic Sections and Formulas
All conic sections originate from the same general equation:
\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\]
where \(A, B, C, D, E, \text{ and } F\) are constants. Which conic section is described depends on the value of each constant and whether it is positive or negative. But this equation is not easy to work with or graph from, so it is not used often.
Each conic section has its own formula, or an equation, that can be used to graph it on a Cartesian coordinate plane like in the images above. Each of the equations below is the standard form or conic form of the equation. These forms of the equation are the most helpful when it comes to graphing and identifying the important features of each conic section.
Circle
As mentioned above, you only need a circle's center and radius in order to write the equation or make the graph.
The equation for a circle with center \((h,k)\) and radius \(r\) is
\[(x-h)^2+(y-k)^2=r^2 \]
or
\[ \frac{(x-h)^2}{r^2}+\frac{(y-k)^2}{r^2}=1.\]
The first form of the equation is likely the one you will see most often. The second form shows how it is related to the ellipse equation shown in the next section.
Ellipse
Ellipses are also rather simple. You only need a center and the distances between the center and the end of each axis.
The equation for an ellipse with center \((h,k)\), major axis \(2a\) and minor axis \(2b\) is
\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1. \]
If \(a>b\) then the ellipse is wider than it is tall, and is called a horizontal ellipse. If \(a<b\) then the ellipse is taller than it is wide, and it is called a vertical ellipse.
For a circle, \(a=b\).
Parabola
You have likely learned a lot about parabolas before. The equation can be in the general form, factored form, or vertex form. The equation below is the conic form that relates a parabola to its important conic section features.
The equation of a parabola with vertex \((h,k)\) and distance \(p\) between the vertex and the focus (or between the vertex and the directrix) is
\[(x-h)^2=4p(y-k)\]
for a parabola that opens up or down, or
\[(y-k)^2=4p(x-h)\]
for a parabola that opens left or right.
Hyperbola
A hyperbola looks like a matching pair of parabolas going outward in opposite directions. For hyperbolas, the center \((h,k)\) is located equidistant from the two vertices of the branches. The distance between the vertices of the two branches is called the transverse axis and is defined as \(2a\). The conjugate axis is perpendicular to this and is defined as \(2b\). This axis helps define how wide open the branches are.
The equation for a hyperbola with center \((h,k)\), transverse axis \(2a\) and conjugate axis \(2b\) is
\[\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \]
for a hyperbola that opens left and right, or
\[\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\]
for a hyperbola that opens up and down.
Rules in Calculating Conic Sections
There are also rules and formulas for finding the important features of different conic sections, like the focus, directrix, and eccentricity. Everything you need for circles and parabolas is built in to those equations above. But for ellipses and hyperbolas, finding these features requires a little extra work.
Finding the Foci and Directrices for Ellipses
For the ellipse equation
\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,\]
which has a horizontal major axis, the formula for finding the distance \(c\) between the center and either focus is
\[c=\sqrt{a^2-b^2}.\]
Once you find this distance, add it to \(h\), the \(x\)-coordinate of the center, to find one focus, and subtract it from \(h\) to find the other focus. The coordinates of the foci will be \((h\pm c,k)\). The foci should always fall inside the ellipse.
The formula for finding the eccentricity \(e\) of an ellipse is
\[e=\sqrt{1-\frac{b^2}{a^2}}\]
which you will need in order to find the directrices. Once you have found \(e\), you can use it in the formula to find the distance \(d\) between the center and the directrices, which is
\[d=\frac{a}{e}.\]
Just like with the focus, once you find this distance, add it to \(h\), the \(x\)-coordinate of the center, to find one directrix, and subtract it from \(h\) to find the other directrix. The equations of the directrix lines will be \(x=h\pm d\). The directrices should always fall outside the ellipse.
If the ellipse is in the form
\[\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1,\]
which has a vertical major axis, the formulas will remain the same except that the distances will be added to \(k\), the \(y\)-coordinate of the center (instead of the \(x\)-coordinate). The directrix lines will be \(y=k\pm d\).
The example below shows how to find the foci and directrices for the ellipse in the graph above.
The graph above is of the ellipse defined by the equation
\[\frac{(x+2)^2}{4}+\frac{(y-1)^2}{1}=1.\]
Find the foci and the directrices.
Foci: Let's start by finding the focal distance from the center using the formula \(c=\sqrt{a^2-b^2}\) and substituting.
\[\begin{align} c&=\sqrt{a^2-b^2}\\c&=\sqrt{4-1}\\&=\sqrt{3}.\\ \end{align}\]
Because this is an ellipse with a horizontal orientation (the major axis is horizontal), you will find the foci to the left and right of the center. The center is at the point (-2,1) (recall that this point is shown in the equation for the ellipse). So one focus will be at \((-2+\sqrt{3},1)\) (or about \((-0.27,1)\)), and the other focus will be at \((-2-\sqrt{3},1)\) (or about \((-3.73,1)\)).
Directrices: In order to find the directrices, you need to find the eccentricity first, using the formula
\[e=\sqrt{1-\frac{b^2}{a^2}}\]
and substituting.
\[\begin{align} e&=\sqrt{1-\frac{b^2}{a^2}}\\&=\sqrt{1-\frac{1}{4}}\\&=\sqrt{\frac{3}{4}}\\&=\frac{\sqrt{3}}{2}.\\ \end{align}\]
Then find the distance between the center and each directrix with the formula \(d=\dfrac{a}{e}\). So
\[d=\frac{2}{\frac{\sqrt{3}}{2}}=\frac{4}{\sqrt{3}}.\]
Add this value to \(h\) to find the equations of the directrix lines. One directrix will be located at \(x=-2+\dfrac{4}{\sqrt{3}}\approx 0.31\), and the other directrix will be located at \(x=-2-\dfrac{4}{\sqrt{3}}\approx -4.31.\)
Take a moment to scroll back up to the ellipse graph and see that these values match up with the foci and directrices indicated on the graph.
Finding the Foci and Directrices for Hyperbolas
For the hyperbola equation
\[\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1,\]
which opens to the left and right (and has a horizontal transverse axis), the formula for finding the distance \(c\) between the center and either focus is
\[c=\sqrt{a^2+b^2}.\]
Once you find this distance, add it to \(h\), the \(x\)-coordinate of the center, to find one focus, and subtract it from \(h\) to find the other focus, just like with an ellipse. The coordinates of the foci will be \((h\pm c,k).\) The foci should always fall inside the branches of the hyperbola.
To find the directrices, use the formula to find the distance \(d\) between the center and the directrices, which is
\[d=\frac{a^2}{c}.\]
Once again, after you find this distance, add it to \(h\), the \(x\)-coordinate of the center, to find one directrix, and subtract it from \(h\) to find the other directrix. The equations of the directrix lines will be \(x=h\pm d\). The directrices should always fall between the branches of the hyperbola.
The example below shows how to find the foci and directrices for the hyperbola in the graph above.
The graph above is of the hyperbola defined by the equation
\[\frac{(x-8)^2}{16}-\frac{(y-6)^2}{9}=1.\]
Find the foci and the directrices.
Foci: Let's start by finding the focal distance from the center using the formula \(c=\sqrt{a^2+b^2}\) and substituting.
\[\begin{align} c&=\sqrt{a^2+b^2}\\&=\sqrt{16+9}\\&=\sqrt{25}=5.\\ \end{align}\]
Because this is a hyperbola that opens to the left and right, you will find the foci to the left and right of the center. The center is at the point (8,6). So one focus will be at \((8+5,6)\text{ or }(13,6)\), and the other focus will be at \((8-5,6)=(3,6).\)
Directrices: Then use the formula \(d=\dfrac{a^2}{c}\) to find the distance between the center and directrices.
\[\begin{align} d&=\frac{a^2}{c}\\&=\frac{16}{5}\\&=3.2.\\ \end{align}\]
Add this value to \(h\) to find the equations of the directrix lines. One directrix will be located at \(x=8+3.2=11.2\), and the other directrix will be located at \(x=8-3.2=4.8.\)
Again, take a moment to scroll up and check the graph to see that this matches.
Conic Sections Examples
There are many different types of conic section problems. The previous examples o finding the foci and directrices are just one type. Below, you will look at another type: how to graph a conic section from its equation.
Graphing an Ellipse
Graphing an ellipse is not too difficult. It is fairly similar to the way you might graph a circle. The example below goes through the steps.
Graph the equation:
\[\frac{(x-1)^2}{4}+\frac{y^2}{16}=1.\]
Step 1: Identify the type of conic section and the orientation.
When you see a conic section equation with both an \(x^2\) term and a \(y^2\) term, you should automatically think of either an ellipse or a hyperbola. The only meaningful difference in the equations is the addition or subtraction sign between the fractions. Ellipses have addition between the terms, like this one.
The orientation is determined by where the major axis is. Here \(a=2\) and \(b=4\), so \(a<b\) and this ellipse has a "vertical" orientation.
Step 2: Plot the center and the major and minor axes.
All the information you need for these things is right there in the equation. The values in the parentheses with the variables make the center. Recall that the standard form of the ellipse equation includes a subtraction sign in front of the coordinate. For this equation, the center is \((h,k)=(1,0)\).
The denominators of the terms tell you the distance from the center point to the end of each axis. The major axis (here, the vertical axis) is \(2a=8\), or \(a=4\), and the minor axis (here, the horizontal axis) is \(2b=4\), or \(b=2\).
Plot these on a graph, like in the image below.
Fig. 11 - Sketch of the center, major axis, and minor axis needed to sketch the graph of an ellipse.
Step 3: Connect the endpoints of the axes to form an ellipse.
Start at any endpoint of either axis. Sketch a curve to connect it to an adjacent endpoint of an axis. Keep going until the ellipse is closed. See the image below for what the final graph should look like.
Fig. 12 - Graph of the elliptical equation \(\frac{(x-1)^2}{4}+\frac{y^2}{16}=1.\)
Let's take a look at hyperbolas next.
Graphing a Hyperbola
Graphing a hyperbola from its equation can be tricky. The example below walks through the steps necessary.
Graph the equation:
\[\frac{(y-4)^2}{25}-\frac{(x+2)^2}{9}=1.\]
Step 1: Identify the type of conic section and the orientation.
Notice the subtraction sign between the fractions. That indicates that this is the equation for a hyperbola. To check its orientation, i.e., whether it opens left/right versus up/down, check to see which variable has the negative sign. Because the \(x\) has the negative sign in front of its parentheses, then this hyperbola will be vertically oriented and open up and down.
Step 2: Use the equation to identify the center and vertices.
The center is part of the equation, so the center \((h,k)=(-2,4)\). The denominator of the first fraction in the equations tells you how far away the vertices of the branches of the hyperbola are from the center. In the standard equation, the denominator is defined as \(a^2\). So for this hyperbola, \(a=5\). And because the hyperbola is oriented up and down, the vertices will be \(5\) units above and below the center, so \((-2,9)\) and \((-2,-1)\). The line segment between the vertices, which has the length \(2a\), is the transverse axis.
Step 3: Plot the important points, axes, and asymptotes.
You already know the transverse axis from Step 2. You will also need to know the conjugate axis, which you can find from the denominator of the second fraction. That denominator is \(b^2\) in the equation, so for this equation \(b=3\). The conjugate axis has a length of \(2b=6\), so \(b=3\), and is perpendicular to the transverse axis through the center.
Next, you need the asymptotes that form the boundaries of the hyperbola. Use the transverse and conjugate axes to sketch a rectangle. Then sketch the diagonals of the rectangle, extending them beyond the rectangle itself, to form the asymptotes. Check the diagram below to see what should be sketched out so far.
Fig. 13 - Sketch of the center, vertices, transverse and conjugate axes, and asymptotes needed to sketch the graph of a hyperbola.
Step 3: Sketch the branches.
Starting at each vertex, sketch a u-shaped curve that approaches each asymptote. It should look like the graph below.
Fig. 14 - Graph of the hyperbolic equation \(\frac{(y-4)^2}{25}-\frac{(x+2)^2}{9}=1.\)
Conic Sections - Key takeaways
- Conic Sections are the result of an intersection of a double-cone with a plane.
- There are four conic sections: circle, ellipse, parabola, and hyperbola.
- Each conic section has a focus and directrix (or two of each) that determine the eccentricity, or curvature, of the conic section.
- The standard form of the equation for each conic section is:
- Circle: The equation for a circle with center \((h,k)\) and raduis \(r\) is \[(x-h)^2+(y-k)^2=r^2\]
- Ellipse: The equation for an ellipse with center \((h,k)\), major axis \(2a\) and minor axis \(2b\) is \[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1. \] If \(a>b\) then the ellipse is wider than it is tall, and is called a horizontal ellipse. If \(a<b\) then the ellipse is taller than it is wide, and it is called a vertical ellipse.
- Parabola: The equation of a parabola with vertex \((h,k)\) and distance \(p\) between the vertex and the focus (or between the vertex and the directrix) is \[(x-h)^2=4p(y-k)\]for a parabola that opens up or down, or\[(y-k)^2=4p(x-h)\] for a parabola that opens left or right.
- Hyperbola: The equation for a hyperbola with center \((h,k)\), transverse axis \(2a\) and conjugate axis \(2b\) is \[\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \] for a hyperbola that opens left and right, or \[\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\] for a hyperbola that opens up and down.
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