What are conic sections?


The conic sections are a class of curves, some closed (like circles) and some open (like a parabola), that are formed by taking "slices" of right-regular cones. They are

  • Circle – slice parallel to the cone base
  • Ellipse – slice not parallel to the cone base and not cutting through the base, and
  • Hyperbola – slice parallel to the cone axis (the line from the tip through the center of the base).
  • Parabola – slice parallel to the cone angle

Conic sections are a subsection of the bigger topic of analytic geometry or coordinate geometry.

Just to refresh your memory, a right-regular cone is formed by revolving a right triangle around one of it's sides so that it "sweeps out" the shape of a cone.

The triangle on the right has only been rotated through 270˚, or ¾ of a circle, so that you can see what's going on. The cone has an axis of symmetry through its center, a circular base and a slant angle that reflects the sharpness of its point.


1. The circle

The circle is a closed figure formed by the intersection of the surface of a right-regular cone by a plane parallel to the base of the cone. A circle is actually just a special case of the ellipse, which we'll get to below.

The equation of the simplest circle, one centered at the origin with radius r = 1 is :

x2 + y2 = 1

In general, the formula for a circle centered at the orign is:

x2 + y2 = r2

The circle on the right shows how that equation works. Any point on the circle lies (by definition) a distance r from the center. The coordinates of that point (x, y) and the length r are related by the Pythagorean theorem. Two such points are shown with their x, y coordinates.

Convince yourself that the Pythagorean theorem is true for the point in the lower left quadrant, too, and further, that it must be true for any point on the circle.

We can use our function transformations to shift the center of a circle from left to right and up & down. The circle equation then refines to:

(x - h)2 + (y - k)2 = r2

where the point (h, k) is the center; h and k are just horizontal and vertical translation parameters, respectively, analogous to those we used in our study of functions.

The circle

A circle is the set (sometimes called the "locus") of all points equidistant from a single point called the center. A circle of radius r, with center at (h, k) is described by

(x - h)2 + (y - k)2 = r2

X

Locus

A curve or other figure formed by all points satisfying a particular equation.


Practice problems

Each of the expressions below is the equation of a circle. Find the location of the center of the circle and its radius.

For the last two, try completing the square on the x- and y-terms. Group the terms containing x together on the left, the terms containing y together on the left, and the constants on the right. Then complete the square on the x and y-groups, accumulating the additional constants on the right. reduce to the standard form of the circle, then just read off your answers.

Roll-over each problem to see the answer. The complete solutions are also available as a .pdf file.

1. Find the center and radius of the following circles

(a) x2 + y2 - 12 = 0

(b) x2 + y2 + 12 = 0

(c) x2 + y2 - 8y = -13

(d) x2 + y2 + 4(x - y) = 17

(e) x (x - 2) + y2 = 80

(f) x2 + y2 + 10(x + y) = -25

2. Sketch graphs of 2x - y = 7 and x2 + y2 = 7. Find the coordinates of intersection by solving the equations simultaneously. Note: The graphs may be tangent or fail to intersect.

3. Sketch graphs of y = x√3 and x2 + (y - 4)2 = 16. Find the coordinates of intersection by solvign the equations simultaneously. Note: The graphs may be tangent or fail to intersect.

4. Find the length of a tangent line segment from (10, 5) to the circle x2 + y2 = 25.

5. Sketch the graph of (x - 3)2 + (y - 4)2 ≤ 25

6. Write the equation of the circle described:

(a) The center is (2, 3) and the circle passes through (5, 6)

(b) The center is (-3, 1) and the circle is tangent to x = 4

(c) The circle is tangent to the x-axis at (4, 0) and has y-intercepts -2 and -8.

(d) The circle contains (-2, 16) and has x-intercepts x = -2 and x = -32.


2. The ellipse

An ellipse is the intersection of the surface of a right-regular cone with a plane so that the plane doesn't intersect the bottom of the cone. The result is a smooth, closed curve, like a circele. In fact, a circle is just a special kind of ellipse. The ellipse is a very important curve in astrophysics; all orbits of celestial bodies are elliptical.

The equation of an ellipse follows directly from the equation of the circle above. Simply think of an ellipse as a circle with two different radii. The figure below will help you see it.


ellipse conic



Content on this page requires a newer version of Adobe Flash Player.

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← The animation illustrates one handy way to make an ellipse. Imagine putting two tacks in a board. Now loop a string (red in the animation) around the tacks and hold it in a taut triangle with a pencil. Holding the string taut with a pencil, trace out the figure.

In an ellipse, the sum of the distance of any point on the curve to each focus is constant (just like the length of the loop of string stays constant). Note that the part of the "string" between the tacks is always the same, so we can ignore that part.

This is a Flash animation. If your browser or mobile device doesn't play it, I'm sorry. I'll eventually convert it to an html5 animation!


Anatomy of an ellipse

The major axis of an ellipse is 2a units long and the minor axis is 2b units long. The sum of the distances d1 and d2 at any point on the ellipse is 2a, and the distance between the center of the ellipse and either focus is c = (a2 - b2)1/2.

The length a always refers to the major axis. If the major axis lies along the y-axis, a and b are swapped in the equation of an ellipse (below).

Ellipses have two axes of symmetry. A longer, narrower ellipse is said to be more eccentric or to have a larger eccentricity. Note that the equation below reverts to a circle in the special case that a=b.

The Ellipse

The equation of an ellipse centered at (0, 0) with major axis a and minor axis b (a > b) is

If we add translation to a new center located at (h, k), the equation is:

The locations of the foci are (-c, 0) and (c, 0) if the ellipse is longer in the x direction, and (0, -c) & (0, c) if it's elongated in the y-direction. c2 = a2 - b2.

Example 1: Sketching an ellipse using "the box"

(which will be handy with hyperbolas, too)

Here's an example of an ellipse, the graph of which we might want to sketch:

The first thing we can do is just read off the coordinates of the vertext, (2, -2). These are just transformations (translations) of the figure along the x- and y-axes, respectively. Remember that we always subtract the translation from the variable of interest, so (x - 2)2 in the denominator means "translate 2 units to the right." If it was (x + 2)2, well that's really (x - (-2))2, or a translation of 2 units to the left.

Now we can define a "box" in which the ellipes lives. It's 6 units wide (3 units, or the root of 9, from the center in each x-direction), and 4 units tall (4 units, or the root of 16, from the center in each y-direction).

To sketch the ellipse we begin with the box, with the appropriate center drawn in:

The box extends ±3 units in the x-direction and ±4 units in the y-direction, as the equation suggests. We generally call the largest radius a and the smallest b, but it's really not necessary to remember that if you can just remember that one is associated with x and one with y in the equation. Just follow what the equation tells you.

Now we can calculate the location of the foci:

The foci always lie along the long axis of the ellipse, and in this case they're √5 units above and below it. With all of this information in hand, we can fully draw the ellipse. It fits inside the box and we can label the foci and each of the four vertices.


Practice problems


1. Find the coordinates of the center, vertices and foci of these ellipses:

2. Each of these ellipses is centered at the origin. Find the equation of each:

(a) vertex (7, 0), minor axis is 2 units long (total length).

(b) vertex (0, -13), focus (0, -5)

(c) vertex (0, -9), minor axis 6 units long.

3. Sketch the graphs of 9x2 + 2y2 = 18 and 3x + y = -3 on a graph, then determine the points of intersection algebraically.


3. The hyperbola


A hyperbola is formed from the intersection of a plane with a right-regular cone so that the plane is parallel to the axis of the cone (left). Hyperbolae (the plural) always come in pairs of two open curves, formed from the intersection of the plane with two cones, as shown. A hyperbola can be thought of as an ellipse turned inside-out.

Hyperbolae have many important applications in science, math and engineering. You might have seen hyperboloid cooling towers of power plants, often huge and visible for miles around.

cooling towers

Image: intmath.com

These towers are particularly good at creating upward air flow. Cooler air is pulled naturally in at the bottom by the difference in air pressure between the top and the bottom of the towers. Rising steam, usually produced from hot water generated in power plants, is cooled rapidly and condenses to form billowing clouds.

We can think of a hyperbola as an ellipse turned inside-out. All that's necessary to convert an ellipse into a hyperbola is to change the addtion in the equation to subtraction. Here is a look at the anatomy of a hyperbola:

Anatomy of a hyperbola

A hyperbola is the collection of all points that meet this condition: The difference of the distances from any point, P, on either curve, to the two foci, F1 and F2, is constant. That means that the difference in the lengths of the green lines above is constant. That difference is equal to the distance between the vertices of the two curves, found at (-a, 0) and (a, 0) if the curves open side-to-side, and (0, -a) to (0, a) if it opens up-down.

We can sketch a hyperbola in the same way as we sketch an ellipse from its equation. First draw the box, of dimensions a x b. If the term that contains x is positive, the curves of the hyperbola open to the left and right. If the y is positive, they open upward and downward. Each of the curves has an asymptote defined by the diagonals of the box, and the locations of the foci are outside of the box: c2 = a2 + b2.

The hyperbola

The equation of a hyperbola centered at the origin is

If we translate the center of the hyperbola to (h, k), the equation becomes

The distance to the foci from the center along the major (longest) axis is c, where c2 = a2 + b2.

Example 2: Sketching the graph of a hyperbola

The box method is even more helpful ...

Here's a step-by-step guide to sketching this hyperbola:

The approach is very similar to an ellipse: We identify the center

and then the dimensions of the box. The half-width of the box in the x-direction is 5 = √25 and that in the y-direction is 4 = √16.

So we can draw the box just as we would if the - sign were a + and this was an ellipse:

The asymptotes of the two curves of the hyperbola are the diagonals of the box, and the vertices are along the x-axis (because the y-term is subtracted from the x-term in this example). Finally, we can sketch in the curves of the hyperbola and calculate the positions of the foci using

Here is the final graph:


Practice problems


Sketch graphs of these hyperbolas. Make sure to label vertices, foci, and a, b & c dimensions:


4. The Parabola

A parabola is formed by the intersection of the surface of a right-regular cone and a plane, where the plane is parallel to the slant angle of the cone.

We already know that a parabola is the graph of a quadratic function, and that the simplest parabola is f(x) = x2, with vertex at (0, 0) and a line of symmetry along the y-axis (below)

A parabola has a vertex that is intersected by a line of mirror symmetry.

The parabola can also be described another way, as the locus (set of locations) of all points equidistant (the same distance) from a point called the focus and a line called the directrix.

In the lower figure (left) the focus is labeled F and the directrix D. All pairs of line segments FPi and PiDi , where i = 1, 2, 3, ... , are congruent.

For a parabola with its axis of symmetry parallel to the y-axis (which would make it a function), f(x) = ax2 + bx + c, we have:

focus & directrix of parabola

A useful property of parabolas

An interesting and useful property of parabolas is their ability to focus incoming beams of light (see figure below). As long as the incoming light beams (red), which might be radio waves, visible light or other types of electromagnetic radiation, are nearly parallel, they will all be reflected from the surface of a 2-dimensional parabolic surface (a surface generated by rotating a parabola 180˚ around its axis of symmetry) toward the focus. That's how satellite antennae are able to pick up a small signal from a noisy background. They collect a relatively large "chunk" of incoming signal and focus it onto a small receiver suspended above the parabola. Electronic filters do the rest of the work of separating the signal from the noise.

parabola focusing property

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