Equations of Conic Sections

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.

Conic Section

The curves generated by cutting a cone with a plane is known as conic sections. A cone is made up of two similar conical shapes called nappes. We can acquire a number of shapes depending on the angle of the cut between the plane and the cone and its nappe. We make the following shapes by cutting a cone at various angles with a plane:

• Circle

• Parabola

• Ellipse

• Hyperbola

When a plane intersects a cone at a right angle, the result is an ellipse. The circle is an ellipse with the cutting plane parallel to the base of the cone.

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

Conic Section Formula

The Standard Formula of Conic Sections

The standard forms of a circle, parabola, ellipse, and hyperbola are represented by conic section formulas. The typical form of ellipses and hyperbolas has the x-axis as the major axis and the origin (0,0) as the centre. The vertices are (a, 0) and the foci are (c, 0), and the equations c2= a2 + b2 for an ellipse and c2 = a2 + b2 for a hyperbola define it. Because c = 0 for a circle, a2 Equals b2. The typical form of the parabola has the focus on the x-axis at (a, 0), and the directrix is the line with the equation x = a.

x2+y2=a2 for circle

When a>0, the parabola is y2= 4ax.

x2/a2 + y2/b2 = 1 for  Ellipse

x2/a2 – y2/b2 = 1 for Hyperbola

Conclusion

The intersection of a plane and a double right circular cone is known as a conic section. We can make several sorts of conics by adjusting the angle and location of the intersection. Circles, ellipses, hyperbolas, and parabolas are the four basic types. The intersections will not travel through the cone’s vertices.