What conic section is formed when the intersecting plane is neither perpendicular to the base of a cone nor parallel to the generator?

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A conic section is defined as the curve of the intersection of a plane with a right circular cone of two nappes. There are three types of curves that occur in this way: the parabola, the ellipse, and the hyperbola. The resulting curves depend upon the inclination of the axis of the cone to the cutting plane. The Greek mathematician Apollonius studied conic sections geometrically using this concept. In this section, we shall give an analytic definition of a conic section, and as special cases of this definition, we shall obtain the three types of curves.

When considering conic sections geometrically, a cone is thought of as having two nappes extending in both directions. A portion of a right circular cone of two nappes is shown in the figure below.

A generator (or element) of the cone is a line lying in the cone, and all generators of a cone contain the point V, called the vertex of the cone.

In figure a below, we have a cone and a cutting plane which is parallel to one and only one generator of the cone. This conic is a parabola. If the cutting plane is parallel to two generators, this intersects nappes of the cone, and a hyperbola is obtained. An ellipse is obtained if the cutting plane is parallel to no generator, in which case the cutting plane intersects each generator, as shown in figure c.

A special case of the ellipse is a circle, which is obtained if the cutting plane, which intersects each generator, is also perpendicular to the axis of the cone. Degenerate cases of the conic sections include a point, a straight line, and two intersecting straight lines. A point is obtained if the cutting plane contains the vertex of the cone but does not contain a generator. This is a degenerate ellipse. If the cutting plane contains the vertex of the cone and only one generator, then a straight line is obtained, and this is a degenerate parabola. A degenerate hyperbola is obtained when the cutting plane contains the vertex of the cone and two generators, giving us two intersecting straight lines.

A conic section is the intersection of a plane and a double right circular cone . By changing the angle and location of the intersection, we can produce different types of conics. There are four basic types: circles , ellipses , hyperbolas and parabolas . None of the intersections will pass through the vertices of the cone.

If the right circular cone is cut by a plane perpendicular to the axis of the cone, the intersection is a circle. If the plane intersects one of the pieces of the cone and its axis but is not perpendicular to the axis, the intersection will be an ellipse. To generate a parabola, the intersecting plane must be parallel to one side of the cone and it should intersect one piece of the double cone. And finally, to generate a hyperbola the plane intersects both pieces of the cone. For this, the slope of the intersecting plane should be greater than that of the cone.

The general equation for any conic section is

A x 2 + B x y + C y 2 + D x + E y + F = 0 where A , B , C , D , E and F are constants.

As we change the values of some of the constants, the shape of the corresponding conic will also change. It is important to know the differences in the equations to help quickly identify the type of conic that is represented by a given equation.
      If B 2 − 4 A C is less than zero, if a conic exists, it will be either a circle or an ellipse.
      If B 2 − 4 A C equals zero, if a conic exists, it will be a parabola.
      If B 2 − 4 A C is greater than zero, if a conic exists, it will be a hyperbola.

STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:

Circle	( x − h ) 2 + ( y − k ) 2 = r 2	Center is ( h , k ) . Radius is r .
Ellipse with horizontal major axis	( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Ellipse with vertical major axis	( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1	Center is ( h , k ) . Length of major axis is 2 a . Length of minor axis is 2 b . Distance between center and either focus is c with c 2 = a 2 − b 2 , a > b > 0 .
Hyperbola with horizontal transverse axis	( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Hyperbola with vertical transverse axis	( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1	Center is ( h , k ) . Distance between the vertices is 2 a . Distance between the foci is 2 c . c 2 = a 2 + b 2
Parabola with horizontal axis	( y − k ) 2 = 4 p ( x − h ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h + p , k ) . Directrix is the line x = h − p Axis is the line y = k
Parabola with vertical axis	( x − h ) 2 = 4 p ( y − k ) , p ≠ 0	Vertex is ( h , k ) . Focus is ( h , k + p ) . Directrix is the line y = k − p . Axis is the line x = h

Solving Systems of Equations

You must be familiar with solving system of linear equation . Geometrically it gives the point(s) of intersection of two or more straight lines. In a similar way, the solutions of system of quadratic equations would give the points of intersection of two or more conics.

Algebraically a system of quadratic equations can be solved by elimination or substitution just as in the case of linear systems.

Example:

Solve the system of equations.

x 2 + 4 y 2 = 16 x 2 + y 2 = 9

The coefficient of x 2 is the same for both the equations. So, subtract the second equation from the first to eliminate the variable x . You get:

3 y 2 = 7

Solving for y :

3 y 2 3 = 7 3 y 2 = 7 3 y = ± 7 3

Use the value of y to evaluate x .

x 2 + 7 3 = 9 x 2 = 9 − 7 3 = 20 3 x = ± 20 3

Therefore, the solutions are ( + 20 3 , + 7 3 ) , ( + 20 3 , − 7 3 ) , ( − 20 3 , + 7 3 ) and ( − 20 3 , − 7 3 ) .

Now, let us look at it from a geometric point of view.

If you divide both sides of the first equation x 2 + 4 y 2 = 16 by 16 you get x 2 16 + y 2 4 = 1 . That is, it is an ellipse centered at origin with major axis 4 and minor axis 2 . The second equation is a circle centered at origin and has a radius 3 . The circle and the ellipse meet at four different points as shown.