What kind of conic is this

Conic sections are used in many fields of study, particularly to describe shapes. For example, they are used in astronomy to describe the shapes of the orbits of objects in space. They could follow ellipses, parabolas, or hyperbolas, depending on their properties. It can be thought of as a measure of how much the conic section deviates from being circular.

The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. This property can be used as a general definition for conic sections. The eccentricity of a circle is zero. Note that two conic sections are similar identically shaped if and only if they have the same eccentricity.

Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix. In the next figure, each type of conic section is graphed with a focus and directrix. The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix. These are the distances used to find the eccentricity.

Conic sections and their parts: Eccentricity is the ratio between the distance from any point on the conic section to its focus, and the perpendicular distance from that point to the nearest directrix. From the definition of a parabola, the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix.

This means that, in the ratio that defines eccentricity, the numerator is less than the denominator. In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix. This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus. Conic sections are formed by the intersection of a plane with a cone, and their properties depend on how this intersection occurs.

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. Types of conic sections: 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. 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:. 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. 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:. 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. 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. Conic sections graphed by eccentricity: This graph shows an ellipse in red, with an example eccentricity value of [latex]0.

It also shows one of the degenerate hyperbola cases, the straight black line, corresponding to infinite eccentricity. The circle is on the inside of the parabola, which is on the inside of one side of the hyperbola, which has the horizontal line below it.

In this way, increasing eccentricity can be identified with a kind of unfolding or opening up of the conic section.

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

Hyperbolas have two branches, as well as these features:. It is the axis length connecting the two vertices. 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. Privacy Policy. Skip to main content. Conic Sections. Search for:. Introduction to Conic Sections. Length of minor axis is 2 b. Distance between the vertices is 2 a.

Distance between the foci is 2 c. Vertex is h , k. 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. 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:. Use the value of y to evaluate x. 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. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors.

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