Conic sections are the curves formed when a plane cuts a cone, these include circles, ellipses, parabolas and hyperbolas, as shown in below figure. For example, the cross section is a circle if the cone is cut horizontally.
A parabola is the set of all points in the plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex lies halfway between the focus and the directrix, and the axis of symmetry is the line that runs through the focus perpendicular to the directrix, illustrated below.
We first consider parabolas that are situated with the vertex at the origin . If the focus of such a parabola is the point , the axis of symmetry is vertical, and the directrix has the equation . Since any point on the parabola is equidistant from the focus and the directrix, we have
This is the standard equation of a parabola with vertical axis, focus at and vertex at the origin . Interchanging and results in a parabola with horizontal axis, focus at and vertex at the origin, with standard equation
The directrix is the line . The line segment that runs through the focus perpendicular to the axis, with endpoints on the parabola, is called the latus rectum, and its length is the focal diameter of the parabola. Since the distance from an endpoint on the latus rectum to the directrix is , the distance from that endpoint to the focus must be as well, so the focal diameter is .
An ellipse is the set of all points in the plane the sum of whose distances from two fixed points and is a constant. These two points are the foci of the ellipse. Generally, an ellipse is an oval curve that looks like an elongated circle, unless the foci are too close together relative to the length of the string, that is, the sum is much greater than the distance between the foci, then the ellipse will look almost circular:
In fact, if the two foci are the same point, we get a circle, which is a special case of the ellipse. To get the simplest equation of an ellipse and for convenience, we place the foci on the axis at and , with the origin being halfway between them, and let the sum of the distances from a point on the ellipse to the foci to be , that is,
Thus we have
To simplify, we isolate one of the radicals then square both sides:
Repeat the same idea to eliminate the remaining radical:
Notice that to form the ellipse, the sum must be larger than the distance between the foci, that is, , or , thus . The equation is then simplified to
For convenience, we write the standard equation of an ellipse as
where , the foci are at and the center is at the origin. Setting , we get , thus the ellipse crosses the axis at and . These points are called the vertices of the ellipse, and the segment that joins them is called the major axis with length which is equal to the sum of the distances from any point on the ellipse to the foci. Setting , we get , so the ellipse also crosses the axis at and . The segment that joins them is called the minor axis with length .
If the foci of the ellipse are placed on the axis at instead, we get a vertical ellipse with vertices and equation . The graphs of the standard equation of an ellipse and a vertical ellipse are shown below.
As discussed earlier, if is much greater than , the ellipse is almost circular. The eccentricity is the number
which measures the deviation of an ellipse from being circular. Since for every ellipse, the eccentricity satisfies . If is close to , then the ellipse is elongated in shape; if is close to , the ellipse is close to a circle in shape. The below figure demonstrates this idea.
On the other hand, a hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points and is a constant. These points are the foci of the hyperbola. Similarly, to get the simplest equation of a hyperbola, we place the foci at and let the difference of the distances between any point on the hyperbola to the foci to be the positive constant . Thus by definition,
or
which simplifies to
Notice that , or , thus we set to write the standard equation of a hyperbola as
with foci at . Note that by definition and are both positive and . The intercepts , obtained by setting , and the points are the vertices of the hyperbola, similar to an ellipse. However, there is no intercept since there is no solution for . Furthermore, from the equation we see
Thus or . These two parts are called the branches of the hyperbola. The segment joining the two vertices is the transverse axis with length . Similarly, interchanging and leads to a hyperbola with a vertical transverse axis and foci at which has the equation . The graph of each equation are shown below.
The fact that these hyperbolas are symmetric about both the and axes can be verified by the equations. To find the asymptotes, we solve the standard equation of a hyperbola for to get
which shows that as , , confirming the lines shown in the graph are asymptotes of the hyperbola. The rectangle centered at the origin from the above figure is called the central box with sides parallel to the axes, that crosses one axis at and the other at . The asymptotes are then simply the lines obtained by extending the diagonals of the central box.
Asymptotes and the central box are essential aids for graphing a hyperbola. We first plot the points regarding and to construct the central box, then obtain the asymptotes by extending the slopes of the diagonals.
Recall the transformations of functions have the effect of shifting their graphs. Generally, the graph of any equation in and can be shifted in a similar fashion. If and are positive real numbers, how each of the following replacements shift the graph is listed below:
For example, if we shift the graph of an ellipse in standard equation so that its center is at the point , that is, the graph is shifted right and upward, the shifted ellipse has the equation .
