What is formed when a plane intersect a cone parallel to its circular base?

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.

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STANDARD FORMS OF EQUATIONS OF CONIC SECTIONS:
Solving Systems of Equations
Parabola
Hyperbola

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.

Conic sections are a particular type of shape formed by the intersection of a plane and a right circular cone. Depending on the angle between the plane and the cone, four different intersection shapes can be formed. Each shape also has a degenerate form. There is a property of all conic sections called eccentricity, which takes the form of a numerical parameter $e$. The four conic section shapes each have different values of $e$.

This figure shows how the conic sections, in light blue, are the result of a plane intersecting a cone. Image 1 shows a parabola, image 2 shows a circle (bottom) and an ellipse (top), and image 3 shows a hyperbola.

Parabola

A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane. Every parabola has certain features:

A vertex, which is the point at which the curve turns around
A focus, which is a point not on the curve about which the curve bends
An axis of symmetry, which is a line connecting the vertex and the focus which divides the parabola into two equal halves

All parabolas possess an eccentricity value $e=1$. As a direct result of having the same eccentricity, all parabolas are similar, meaning that any parabola can be transformed into any other with a change of position and scaling. The degenerate case of a parabola is when the plane just barely touches the outside surface of the cone, meaning that it is tangent to the cone. This creates a straight line intersection out of the cone's diagonal.

Non-degenerate parabolas can be represented with quadratic functions such as

$f(x) = x^2$

Circle

A circle is formed when the plane is parallel to the base of the cone. Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle. All circles have certain features:

A center point
A radius, which the distance from any point on the circle to the center point

All circles have an eccentricity $e=0$. Thus, like the parabola, all circles are similar and can be transformed into one another. On a coordinate plane, the general form of the equation of the circle is

$(x-h)^2 + (y-k)^2 = r^2$

where $(h,k)$ are the coordinates of the center of the circle, and $r$ is the radius.

The degenerate form of the circle occurs when the plane only intersects the very tip of the cone. This is a single point intersection, or equivalently a circle of zero radius.

This graph shows an ellipse in red, with an example eccentricity value of $0.5$, a parabola in green with the required eccentricity of $1$, and a hyperbola in blue with an example eccentricity of $2$. It also shows one of the degenerate hyperbola cases, the straight black line, corresponding to infinite eccentricity.

Ellipse

When the plane's angle relative to the cone is between the outside surface of the cone and the base of the cone, the resulting intersection is an ellipse. The definition of an ellipse includes being parallel to the base of the cone as well, so all circles are a special case of the ellipse. Ellipses have these features:

A major axis, which is the longest width across the ellipse
A minor axis, which is the shortest width across the ellipse
A center, which is the intersection of the two axes
Two focal points—for any point on the ellipse, the sum of the distances to both focal points is a constant

Ellipses can have a range of eccentricity values: $0 \leq e < 1$. Notice that the value $0$ is included (a circle), but the value $1$ is not included (that would be a parabola). Since there is a range of eccentricity values, not all ellipses are similar. The general form of the equation of an ellipse with major axis parallel to the x-axis is:

$\displaystyle{ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 }$

where $(h,k)$ are the coordinates of the center, $2a$ is the length of the major axis, and $2b$ is the length of the minor axis. If the ellipse has a vertical major axis, the $a$ and $b$ labels will switch places.

The degenerate form of an ellipse is a point, or circle of zero radius, just as it was for the circle.

Hyperbola

A hyperbola is formed when the plane is parallel to the cone's central axis, meaning it intersects both parts of the double cone. Hyperbolas have two branches, as well as these features:

Asymptote lines—these are two linear graphs that the curve of the hyperbola approaches, but never touches
A center, which is the intersection of the asymptotes
Two focal points, around which each of the two branches bend
Two vertices, one for each branch

The general equation for a hyperbola with vertices on a horizontal line is:

$\displaystyle{ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 }$

where $(h,k)$ are the coordinates of the center. Unlike an ellipse, $a$ is not necessarily the larger axis number. It is the axis length connecting the two vertices.

The eccentricity of a hyperbola is restricted to $e > 1$, and has no upper bound. If the eccentricity is allowed to go to the limit of $+\infty$ (positive infinity), the hyperbola becomes one of its degenerate cases—a straight line. The other degenerate case for a hyperbola is to become its two straight-line asymptotes. This happens when the plane intersects the apex of the double cone.