Sections of Cones

The curves formed by the intersection of a plane with a cone are known as conic sections or cone sections. A cone or conic section has three principal sections: parabola, hyperbola, and ellipse (the circle is a special kind of ellipse). The conic portions are made using a cone with two identical nappes.

Conic section

Conic sections are the curves created by cutting a cone with a plane. Nappes are two similar conical shapes that make up a cone. Depending on the angle of the cut between the plane and the cone and its nappe, we can get a variety of shapes. We generate the following shapes by cutting a cone with a plane at various angles:

• Circle

• Parabola

• Ellipse

• Hyperbola

Ellipse is a conic section generated when a plane intersects a cone at a right angle. The circle is a sort of ellipse in which the cutting plane is parallel to the cone’s base.

Parameter of conic section

The conic is defined by three important traits or parameters: focus, directrix, and eccentricity. The circle, ellipse, parabola, and hyperbola are examples of conic figures. And these three key characteristics determine the shape and direction of these shapes. Let’s take a closer look at each of them.

focus

The point(s) around which the conic section is generated is called the focus or foci(plural). Each form of the conic section has its own set of rules. Ellipses and hyperbolas have two foci, while parabolas have one. In the case of an ellipse, the sum of the distances between the two foci is constant. The foci of a circle, which is a particular case of an ellipse, are at the same location, and the distance between all locations and the focus is constant. The parabola is a limiting case of an ellipse with two foci, one at a distance from the vertex and the other at infinity.

Directrix

The conic sections are defined by the directrix line. The directrix is a line drawn parallel to the referred conic’s axis. The ratio of the distance between the directrix and the foci defines every point on the conic. The conjugate axis and the conic’s latus rectum are parallel to the directrix. A circle has no directrix. The parabola has one directrix, while the ellipse and hyperbola each have two.

Eccentricity

The constant ratio of the distance of the point on the conic section from the focus and directrix is the eccentricity of a conic section. Eccentricity is a term used to describe a conic section’s shape. It is a real number that is not negative. The letter “e” stands for eccentricity. 

Circle

The circle is a sort of ellipse in which the cutting plane is parallel to the cone’s base. The circle’s focus is known as the circle’s centre. The radius of the circle is the distance between the locus of the points on the circle and the focus or centre of the circle. For a circle, the value of eccentricity(e) is e = 0. There is no directrix in Circle. The general version of the equation for a circle with a radius of r and a centre at (h, k):

(x−h)2 + (y−k)2 = r2

Parabola

A parabola is a conic section formed when the intersecting plane is at an angle to the cone’s surface. It’s a conic section with a U shape. The eccentricity(e) value for a parabola is e = 1. The intersection of a cone with a plane parallel to its side produces an asymmetrical open plane curve. A parabola is a line-symmetric curve that has the same shape as the graph of y = x2. A parabola’s graph can either open upwards, as in y = x2, or downwards, as in y = – x2. Under the effect of gravity, a projectile’s journey should ideally follow a curve of this shape.

Ellipse

Ellipse is a conic section generated when a plane intersects a cone at a right angle. Ellipse features two foci, as well as a major and minor axis. For an ellipse, the eccentricity(e) value is e 1. Ellipse has two axes of rotation. The general form of an elliptical equation with the centre at (h, k) and the major and minor axis lengths of ‘2a’ and ‘2b’, respectively. The ellipse’s primary axis is parallel to the x-axis. For an ellipse, the conic section formula is as follows.

(x−h)2/a2 + (y−k)2/b2 = 1

Hyperbola

When the fascinating plane is parallel to the axis of the cone and intersects both nappes of the double cone, a hyperbola is created. For hyperbola, the eccentricity(e) value is e > 1. Branches are the two unconnected parts of the hyperbola. Their diagonally opposing arms approach the line’s boundary, and they are mirror reflections of each other.

A hyperbola is a conic section that may be drawn on a plane and intersects a double cone made up of two nappes. The general form of the hyperbola equation with (h, k) as the centre is as follows:

(x−h)2/a2 – (y−k)2/b2 = 1

Conclusion

A conic section (or simply conic) is a curve formed by the intersection of the surface of a cone with a plane in mathematics. The hyperbola, parabola, and ellipse are the three forms of conic sections; the circle is a special case of the ellipse, though it was sometimes considered a fourth type. Conic sections were explored by ancient Greek mathematicians, culminating in Apollonius of Perga’s methodical work on their characteristics around 200 BC.