Conic Section

A conic section, also referred to as just conic, is what results when the surface of a cone crosses with a plane. In the field of mathematics, this type of curve is known as a conic curve. However, the circle is a specific instance of the ellipse and has been referred to as a fourth kind in some circles. Historically, the hyperbola, the parabola, and the ellipse were believed to be the three forms of conic sections. Ancient Greek mathematicians were responsible for the investigation of conic sections, which reached its pinnacle around the year 200 BC with Apollonius of Perga’s methodical work on the properties of conic sections.

 A non-circular conic can be defined as the set of points whose distances from a given point, known as a focus, and a certain line, known as a directrix, are in a fixed ratio, known as the eccentricity, to one another. This is one of the properties that constitute a non-circular conic. A non-circular conic can also be defined as the set of points whose distances from a particular point, which is referred to as a focus, and one particular line, which is referred to as a directrix, are in a fixed ratio, which is referred to as the. The function that determines the type of conic is called the eccentricity, and it is a property of the conic. A conic is a plane algebraic curve of degree 2, and its definition in analytic geometry describes it as the collection of points whose coordinates satisfy a quadratic equation in two variables that can be represented in matrix form. The derivation and algebraic formulation of the geometric properties of conic sections can be done in a straightforward manner using this equation, which facilitates the derivation.

Definition

The term “conic” refers to a curve that is produced when a plane, also known as the cutting plane, intersects with the surface of a double cone, also known as the intersection plane (a cone with two nappes). However, this is not essential; any double cone with a circular cross-section will suffice in this scenario. For the purpose of convenience, it is generally believed that the cone is a right circular cone; however, this is not required. Planes that traverse the vertex of the cone will produce either a point, a line, or a pair of lines that intersect with one another as their end result. Planes that travel through the cone’s apex will result in an intersection with the cone. 

Conics can be broken down into three distinct categories, which are the ellipse, the parabola, and the hyperbola. Even though the circle is a specific subtype of the ellipse, the ancient Greek mathematician Apollonius regarded it to be a distinct fourth category of shape. When a closed curve meets the point at which a cone and a plane cross, the resulting shape is called an ellipse. In order to generate a circle, the cutting plane of the cone must be perpendicular to the axis of the cone. This condition is met when the cutting plane of the cone is parallel to the plane of the generating circle of the cone. If the cutting plane is perpendicular to precisely one of the cone’s generating lines, then the cone is referred to as an unbounded parabolic conic and is considered to be unbounded. In the last case, the figure is a hyperbola, which indicates that the plane touches both halves of the cone, producing two independent unbounded curves in each half. This is the case because the plane contacts both parts of the cone.

It is also possible to define a conic section solely in terms of plane geometry. This definition states that a conic section is the locus of all points P whose distance from a given point F (also known as the focus) is a constant multiple (also known as the eccentricity e) of the distance from P to a fixed line L (called the directrix). This is an alternative way of defining a conic section. In the scenario where e is equal to one, we get a parabola, and when e is greater than one, we get a hyperbola. We get an ellipse when e is less than one.

A circle is an exceptional case that does not possess a focus or directrix on the Euclidean plane, and as a result, it is not defined by either of those features. The eccentricity of a circle is known to be equal to zero, and its focus is known to be located at the circle’s centre; however, the directrix of a circle can only be defined as the line that extends to infinity in the projective plane. 

General equation of conic section

The following are the equations for conic sections:

Circle

When the centre of the equation is at the origin, the equation is written as (0, 0)

When the centre is in the equation (h, k)

(x – h)2 + (y – k)2  =r2

,where r is the radius of the circle

(h,k) is center

Ellipse

(x−h)2/a2 + (y−k)2/b2 = 1

Hyperbola

(x−h)2/a2 – (y−k)2/b2 = 1

Parabola

y2 = 4ax, where a denotes the distance between the origin and the focal point.

Conclusion

Conic section formulas represent the standard forms of a circle, parabola, ellipse, hyperbola. For ellipses and hyperbolas, the standard form has the x-axis as the principal axis and the origin (0,0) as the centre. The vertices are (±a, 0) and the foci (±c, 0)., and are defined by the equations c2= a2 − b2 for an ellipse and c2 = a2 + b2 for a hyperbola. For a circle, c = 0 so a2 = b2. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix is the line with 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