Conic Sections | Research & Encyclopedia Articles

Imagine there are two cone-shaped paper drinking cups, each fastened to the other at its point, or vertex. The figure that would result is described mathematically as a right circular cone (sometimes called a double cone), which is formed by a straight line that moves around the circumference of a circle while passing through a fixed point (the vertex) that is not in the plane of the circle.

If a right circular cone is cut, or intersected, by a plane at different locations, the intersections form a family of plane curves called conic sections (see the figure). If the intersecting plane is parallel to the base of the cone, the intersection is a circle—which shrinks to a point when the plane has moved toward the cone's tip and finally passes through the vertex. If the intersecting plane is not parallel to the base, passes through only one half of the cone, and is not parallel to the side of the cone, then the intersection is an ellipse. If the plane intersects both halves of the cone, and is not parallel

EQUATIONS FOR CONIC SECTION CURVES
Curve	General Equation	Notes	Example
Circle	(x-h)2 + (y-k)2 = r2	center of circle = (h, k) radius = r	x2 + y2 = 49
Ellipse		length of major axis = 2a foci at c and –c b2 = a2 –c2center = (h, k)	x2 + 25y2 = 49
Hyperbola		foci at c and –c a2 + b2 = c2 center = (h, k) equation of asymptotes: y = ±(b/a)x
Parabola	a(x-h)2 + k = y	axis of symmetry: x = h vertex = (h, k)	(x-7)2 + 1 = y

to the side of the cone, then the intersection is a curve that has two branches, called a hyperbola. If the plane intersects the cone so that the plane is parallel to the side of the cone, then the intersection is a curve called a parabola. The equations for the conic section curves can have the general forms summarized in the table.

These conic section curves—the circle, ellipse, hyperbola, and parabola—have been known and named for more than 2,000 years, and they occur in many applications. The path of a thrown ball, the arc of cables that support a bridge, and the arc of a fountain are examples of parabolas. The shape of a sonic boom, the path of comets, and the LORAN navigation system for ships involve the hyperbola. The paths traveled by the planets, dome-shaped ceilings, and the location of the listening points in a whispering gallery involve the ellipse. Of course, examples of the most familiar conic section—the circle—can be found everywhere: the shape of flowers, ripples in a pool, and water-worn stones.

For every point on an ellipse, the sum of the distances from that point to two points called the foci is the same. These two points, the foci, are on the major (longest) axis of the ellipse. The value of this constant sum is 2a.

For every point on a hyperbola, the absolute value of the difference of the distances from that point to two points called the foci is the same. The absolute value of this constant difference is 2a.

Locus.

Jacobs, Harold R. Mathematics, A Human Endeavor. San Francisco: W.H. Freeman and Company, 1970.

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