Conic sections or sections of a cone are the curves obtained by the intersection of a plane and cone. There are three major sections of a cone or conic sections: parabola, hyperbola, and ellipse(the circle is a special kind of ellipse). A cone with two identical nappes is used to produce the conic sections.
All the sections of a cone or conic sections have different shapes but they do share some common properties which we will read in the following sections. Let us check the conic section formulas, conic equations and its parameters, with examples
What is a Conic section?
In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.
The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type.
The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga’s systematic work on their properties.
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.
One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity.
The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables, which may be written in matrix form.
This equation allows deducing and expressing algebraically the geometric properties of conic sections. In the Euclidean plane, the three types of conic sections appear quite different but share many properties.
By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity.
A further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.
Key Takeaway:
- Ellipse is the locus of a point P which moves such that the ratio of its distance from the fixed-point F to its distance from a fixed line is a constant and is always less than 1.
- Parabola is the locus of a point Q which moves such that the ratio of its distance from the fixed-point F to its distance from the fixed-line is a constant and is always equal to 1.
- Hyperbola is the locus of a point R which moves such that the ratio of its distance from the fixed-point F to its distance from the fixed-line is a constant and is always greater than 1.
Definition
A conic is a curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).
It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Planes that pass through the vertex of the cone will intersect the cone in a point, a line, or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, “conic” in this article will refer to a non-degenerate conic.
There are three types of conics: the ellipse, parabola, and hyperbola. The circle is a special kind of ellipse, although historically Apollonius is considered as a fourth type. Ellipses arise when the intersection of the cone and plane is a closed curve.
The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone; for a right cone, this means the cutting plane is perpendicular to the axis.
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.
Equation, Eccentricity, Latus Rectum, Focal Parameter
Focus, Eccentricity, and Directrix of Conic
A conic section can also be described as the locus of a point P moving in the plane of a fixed point F known as focus (F) and a fixed line d known as directrix (with the focus not on d) in such a way that the ratio of the distance of point P from focus F to its distance from d is a constant e known as eccentricity. Now,
- If eccentricity, e = 0, the conic is a circle
- If 0<e<1, the conic is an ellipse
- If e=1, the conic is a parabola
- And if e>1, it is a hyperbola
So, eccentricity is a measure of the deviation of the ellipse from being circular. Suppose, the angle formed between the surface of the cone and its axis is β and the angle formed between the cutting plane and the axis is α, the eccentricity is;
e = cos α/cos β
Parameters of Conic
Apart from focus, eccentricity and directrix, there are few more parameters defined under conic sections.
- Principal Axis: Line joining the two focal points or foci of ellipse or hyperbola. Its midpoint is the centre of the curve.
- Linear Eccentricity: Distance between the focus and centre of a section.
- Latus Rectum: A chord of section parallel to directrix, which passes through a focus.
- Focal Parameter: Distance from focus to the corresponding directrix.
- Major axis: Chord joining the two vertices. It is the longest chord of an ellipse.
- Minor axis: Shortest chord of an ellipse.
Circle – Conic Section
The circle is a special type of ellipse where the cutting plane is parallel to the base of the cone. The circle has a focus known as the center of the circle.
The locus of the points on the circle have a fixed distance from the focus or center of the circle and is called the radius of the circle. The value of eccentricity(e) for a circle is e = 0. Circle has no directrix.
Parabola – Conic Section
When the intersecting plane is at an angle to the surface of the cone we get a conic section named parabola. It is a U-shaped conic section. The value of eccentricity(e) for parabola is e = 1. It is an asymmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side.
The graph of a quadratic function is a parabola, a line-symmetric curve whose shape is like the graph of y = x2. The graph of a parabola either opens upward like y = x2 or opens downward like the graph of y = – x2. The path of a projectile under the influence of gravity ideally follows a curve of this shape.
Ellipse – Conic Section
An ellipse is a conic section that is formed when a plane intersects with the cone at an angle. The ellipse has 2 foci, a major axis, and a minor axis. The value of eccentricity(e) for ellipse is e < 1. Ellipse has 2 directrices.
The general form of the equation of an ellipse with center at (h, k) and length of the major and minor axes as ‘2a’ and ‘2b’ respectively.
Hyperbola – Conic Section
A hyperbola is formed when the interesting plane is parallel to the axis of the cone, and intersect with both the nappes of the double cone. The value for eccentricity(e) for hyperbola is e > 1.
The two unconnected sections of the hyperbola are called branches. They are mirror images of each other, and their diagonally opposite arms approach the limit to a line.
A hyperbola is an example of a conic section that can be drawn on a plane that intersects a double cone created from two nappes.
Applications
Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton’s law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.
The reflective properties of the conic sections are used in the design of searchlights, radio telescopes, and some optical telescopes. A searchlight uses a parabolic mirror as the reflector, with a bulb at the focus; and a similar construction is used for a parabolic microphone.
The 4.2-meter Herschel optical telescope on La Palma, in the Canary Islands, uses a primary parabolic mirror to reflect light towards a secondary hyperbolic mirror, which reflects it again to a focus behind the first mirror.
FAQs.
What are the 4 types of conic sections?
A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. Study the figures below to see how a conic is geometrically defined.
Is conic sections Algebra 2?
Conic sections are known as quadratic relations (not functions) since their equations are of second order but are not always functions. The four basic conic sections are circle, parabola, ellipse and hyperbola. In Algebra 2, we will concentrate on the circle and the parabola.
Why are they called conic sections?
Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. One nappe is what most people mean by “cone,” having the shape of a party hat.
What are the three basic conic sections?
The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type.
What is the difference between conic and conic section?
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 .