What conic section is formed when the plane intersects both cones to form two unbounded curves?

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.

Answer:

Parabola

Explanation:

If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola: the plane intersects both halves of the cone, producing two separate unbounded curves.