Sections of conics

When a plane slices through a cone, it yields a curve known as a conic section. There are many different forms that we can achieve depending on the angle at which the cone and its nappe are sliced. We may make the following shapes by slicing a cone in different directions with a plane: 

• Circle 
• Parabola 
• Ellipse
• Hyperbola 

It is possible to create ellipses by intersecting the cone at an angle. The conic section in which the cutting plane is parallel to the cone’s base is called a ‘circle’ in this context. In order to make a hyperbola, the plane must connect both nappes of the double cone in a straight line. To describe the shape formed when the intersecting plane passes through the cone’s outer surface at an angle, we call it a ‘parabola.’

Eccentricity, Focus, and Directrix

The conic’s focus, directrix, and eccentricity are three of the essential parameters. There are four basic types of conic shapes: circles, ellipse, parabola, and hyperbola. Their shapes are also influenced by these three traits, which dictate their orientation.

Eccentricity

The eccentricity of a conic section is defined by its constant ratio between its distance from its focus and its directrix. It is only through the application of eccentricity that the shape of a conic section may be uniquely defined.  ‘e’ stands for ‘eccentricity.’ The eccentricity of conic sections is a non-negative real number. If two conic sections have the same eccentricity, they will both be same conic sections. The increasing eccentricity causes the conic section to wander farther from the circle’s shape over time. The value of e for different conic sections is as follows.

• For a circle, e = 0.
• For ellipse, 0 ≤ e < 1
• For parabola, e = 1
• For hyperbola, e > 1   

Focus

A conic section’s foci are the point around which the conic section is constructed. There is a specific definition for each category. Ellipses and hyperbolas have two foci, while parabolas have one. A circle is a specific instance of an ellipse in which the foci are both located at the same position and the distance to all points is constant. A parabola is a limiting case of an ellipse that has one focus at a distance from the vertex and another focus at infinity. Hyperbola also has two foci.

Directrix

Directrix is used to define a conic section. It is the straight line drawn perpendicular to the axis of the conic section. Each point on the conic is determined by the ratio of its distance from the directrix and the foci. It is oriented parallel to the conjugate axis of the conic(in case of hyperbola) and the latus rectum. The parabola has one directrix, the ellipse  and hyperbola has two,while the circle has none.

Conic Section Parameters

Some other parameters of conic sections include those listed above, as well as others like the principal and minor axes, latus rectums, the focal parameter, and more. The Conic section parameters are as follows:

• Principal axis/Major axis: A conic’s principal axis is the line going between its centre and foci, and it is also known as the major axis.
• Conjugate axis/Minor axis: The conjugate axis is the axis drawn perpendicular to the major axis, running through the conic’s centre. Similarly, it is the minor axis of the conjugate axis itself.
• Centre: When the major and conjugate axes of a conic intersect, the intersection point is referred to as the conic’s centre.
• Vertex: Conic vertex refers to the place where conic intersects with the axis.
• Focal Chord: The focal chord of a conic is the chord that passes through the conic section’s focal point. The focal chord divides the conic section into two different sections at each of these places.
• Focal Distance: When a point (x1,y1) on the conic is measured in relation to one of the foci, the focal distance is calculated. Ellipses and hyperbolas have two foci. Thus, we have two focal lengths. Parabola has one focus so it has one focal length.
• Latus Rectum: The focal chord is perpendicular to the conic’s axis. Parabolic latus rectum is LL’ = 4a (for a parabola having form of y2=4ax) in length. Ellipses and hyperbolas have the same Latus Rectum length, which is 2b2/a(where b is semi minor axis and a is semi major axis).
• Tangent: A tangent is a line that touches the conic at a single point. The point of contact is where the tangent meets the conic. Two tangents can also be drawn to the conic from an external point.

• Normal: A normal is a line drawn at an angle perpendicular to the tangent. In a circle, it is also a straight line running through the point of contact and centre. Each of the conic’s tangents can have its own normal.

• Chord of Contact: The chord of contact is the chord drawn to unite the tangent point of touch drawn from an external point to the conic.

• Pole and Polar: To describe a position outside the conic section as a ‘pole,’ we use the term ‘polar’ to describe the locus at which the points of intersection of the tangents, which are drawn at the ends of the chords, meet.
• Auxiliary Circle and Director Circle: The auxiliary circle has a diameter equal to the principal axis of the ellipse. For an ellipse, the auxiliary circle’s equation is written as  x2 + y2 = a2(a is semimajor axis). When the perpendicular tangents are drawn to the ellipse, the locus of the point of intersection of perpendicular tangents is known as the director circle. To determine the director circle’s equation for an ellipse, use the formula x2/a2 + y2/b2 = 1. The equation of director circle is x2 + y2 = a2+b2

• Asymptotes: These are straight lines that are drawn parallel to the hyperbola and presumed to meet at infinity. Hyperbolic asymptotes are defined by the formula y = bx/a and y = -bx/a correspondingly. Hence, there are two asymptotic equations for hyperbolas.

Different Sections of the Cone and Conic Section Formulas

Parabola 

A parabola is a conic section whose intersecting plane is at an angle to the cone’s surface. 

Conic sections in the shape of a U characterise this structure. Its eccentricity (e) is equal to one. It is an open plane curve generated by the intersection of a cone with a plane parallel to the side of the cone Parabolas are line-symmetric curves that look like the graph of the quadratic function. For example, the graph of y = x2 opens upward and y = – x2 opens downward like this. Ideally, a projectile’s trajectory under the effect of gravity is a parabolic path.

Ellipse 

A plane meets the cone at an angle, forming as an ellipse, creating a conic section. The major axis and the minor axis are both present in the ellipse. e<1 is the ellipsoid’s eccentricity value, “2a” and “2b” are two notations used to denote the lengths of two principal axes in general form for ellipses with centres at (h,k). The ellipse’s major axis is perpendicular to the x-axis. 

(x−h)2/a2 + (y−k)2/b2 = 1

Note: If the major axis is parallel to the y-axis, switch the places of a and b in the formula given above.

Circle 

A conic section with a cutting plane that is parallel to the base is known as a circle. In geometry, a circle’s centre is referred to as its ‘centre of gravity.’ The radius is the measure of distance between a point on a circle and its focal point, also known as its centre. For a circle, the value of eccentricity (e) is e = 0. 

There is no axis of rotation for a circle. For a circle with a centre at (h,k), the following is the generic version of the equation:

(x−h)2+ (y−k)2= r2

Hyperbola

In order to form a hyperbola, the intersecting plane must intersect both nappes of the double cone at the same time. For hyperbola, the value of eccentricity (e) is greater than 1. Branches refer to the hyperbola’s two unconnected parts. Their diagonally opposite arms reach the boundary of a line, making them mirror images of one another. 

The general form of the equation of the hyperbola with (h, k) as the centre is as follows.

(x−h)2/a2 – (y−k)2/b2 = 1

Conic Section Standard Forms

This is the usual form for ellipses, which uses the x-axis as the primary axis and the origin (0,0) as the centre. Described by the hyperbolic equation c2 = a2+ b2 which is equivalent to c2= a2 -b2 for an ellipse. So, for a circle, c = 0, a2=b2. The standard version of the parabola has the x-axis focus at (a, 0) and the directrix is the line with the equation x = -a.

• Circle: x2+y2= a2
• Parabola: y2= 4ax when a>0
• Ellipse: x2/a2+ y2/b2= 1
• Hyperbola: x2/a2– y2/b2 = 1

Conclusion

Besides geometry, conic sections can be found in a wide range of fields, including astronomy, optics, and architecture. They have a wide application in the field of science and technology. 

The curve formed when a plane and a cone intersect is referred to as a conic section. Different types of conic sections in mathematics can be found by measuring the angle formed between the plane and the right circular cone’s intersection with the section.