On Conic Sections

Certain geometric figures are called conic sections because they are formed by intersecting a plane with a right circular cone in some certain way. Four conic sections are to be discussed here:

• Circle
• Ellipse
• Parabola
• Hyperbola

The Circle


If we intersect a plane and a cone in such a way that the plane is parallel to the base of the cone, we form our first conic section: the circle.


A circle is the set of all points on a plane that have a common distance from a fixed point. The fixed point is called the center of the circle, and the common distance is called the circle's radius.


When graphed in the xy-plane, the standard form of an equation of a circle with center at (h,k) and radius r is given by 

(x – h)² + (y – k)² = r²

In particular, if the center is at the origin (0,0) the equation is

x² + y² = r²

The Ellipse

Let's go back to the plane-cone intersection. If this time we tilt the plane at some acute angle, then the circle becomes somewhat flattened, and we have our next conic section: the ellipse.

An ellipse is the set of all points on a plane such that the sum of their distances between two fixed points is a constant. The fixed points are called the foci (plural of focus) of the ellipse.

In the xy-plane, the standard form of an equation of an ellipse with center at (h,k) is

(x – h)²/a² + (y – k)²/b² = r²

In particular, if the center is at the origin (0,0), the equation is

x ²/a² + y²/b² = r²

 In both cases, the foci are of distance c from the center, where    c² = a² – b²
 
The major axis (the segment that contains the foci) has length 2a, while the minor axis has length 2b. These two axes intersect at the center.



The Parabola

Let's go back to the plane-cone intersection we have been talking about. This time let us tilt the plane further so that it intersects with the cone's base. Notice that we have formed an open figure this time. This is our third conic section: the parabola.

A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is the directrix. There is one point on the parabola that goes halfway between the focus and the directrix: this is called the vertex. The line that contains the focus and vertex is called the parabola's axis.

Suppose we situate our parabola such that the vertex is at the origin (0,0) and the axis is parallel to the y-axis. Then an equation for such a parabola is

 y = ax²

where a = 1/(4p) and p is the distance from the focus to the parabola or from the parabola to the directrix. If a is positive, the parabola opens upward. If negative, then it opens downward.

Now, if the axis of the parabola is parallel to the x-axis, then the equation becomes

x = ay²

This time, if a is positive the parabola opens to the right. If negative, then it opens to the left.









The Hyperbola


Let's take two right circular cones, place them tip to tip, just like the sides of an hourglass. Suppose we intersect a plane such that it is perpendicular to both bases of the cones. We have formed our fourth and last conic section: the hyperbola. Notice that the hyperbola is actually two parabolas opening in opposite directions from each other.


A hyperbola is the set of all points on a plane such that the difference of their distances from two fixed points is a constant. Notice that this definition is similar to that of the ellipse: only that we change the "sum" into "difference". The two fixed points are likewise called the foci of the hyperbola.


The equation of a hyperbola is the same as that of the ellipse, but we change the plus sign into a minus sign. The hyperbola has two asymptotes,

y = (b/a)x   and   y = -(b/a)x

An asymptote to a curve is a line that the curve almost reaches but never actually touches. It serves as some kind of boundary.

Source: Calculus: Concepts and Contexts 2nd Edition by James Stewart