Study About Equation of Ellipse and Parabola Here

Conic sections are one of the most important topics in Geometry. Furthermore, conic sections have wide applications. For example, the equation of ellipse is crucial in calculating the Earth's orbit around the sun. There are a few types of conic sections in mathematics that can be described depending on the angle formed by the plane and the intersection of the right circular cone with it. Namely;

• Circle
• Ellipse
• Parabola
• Hyperbola

Let’s learn about two very important conic sections (Parabola and Ellipse) from the aforementioned list in detail.

What is Parabola?

A parabola is a section of a right circular cone when it is cut in half by a plane that is parallel to the cone's generator. It is a moving point locus whose distance from the focus is equivalent to its distance from a fixed line (directrix).

Standard Equation of Parabola

The general presentation of the Equation of Parabola is as follows:

When the directrix is parallel to the y-axis, y² = 4ax

If the parabola is sideways, that is, if the directrix is parallel to the x-axis, the standard parabola equation is x² = 4ay

The equations can also be y² = -4ax and x² = -4ay if the parabola is in the negative quadrants in addition to these two.

Properties of Parabola

Important features of a parabola include the following:

• Eccentricity is a conic section parameter that indicates the conic section's roundedness. While less eccentricity suggests greater spherical behaviour, more eccentricity suggests less spherical behaviour. Eccentricity is denoted by the letter ‘e’. The ratio of the focus' distance to the vertex's distance from just one point on the plane determines a parabola's eccentricity. Any parabola consequently has an eccentricity of 1.
• The parabola is symmetric about its axis.
• The axis is perpendicular to the directrix.
• The focus is joined to the vertex by the axis.
• The tangent and directrix are parallel at the vertices.
• The vertex is the midway of the focus, where the directrix and axis connect.
• If an is the distance between the focus and the vertex, then 2a is the distance between the focus and the plane point, which is the same as the distance between the focus and the directrix. The axis of symmetry is located halfway up the latus rectum, showing that the length is 4a and the half is 2a.

What is an Ellipse?

An ellipse is a two-dimensional shape defined along its axes in geometry. When a plane intersects a cone at an angle to the base of the cone, an ellipse is created.

There are two focal points. For all places on the curve, the total of the two distances to the focal point is always constant.

A circle is also an ellipse in which the foci are all at the same location, which is the circle's centre.

Major and Minor Axis

An Ellipse is characterised by its two axes along the x and y axes:

Major axis

Minor Axis

The major axis is the ellipse's longest diameter (typically represented by 'a'),which runs through the centre from one end to the other, at the broadest section of the ellipse. The minor axis is the ellipse's shortest diameter (denoted by 'b'),passing through the centre at its narrowest point.

Properties of an Ellipse

• Ellipse has two focus points, also known as foci.
• The ellipse's directrix is the fixed line parallel to the minor axis at distance (d) from the centre.
• The ellipse's eccentricity spans from 0 to 1.
• The sum of the distances between the locus of an ellipse and its two focal points is constant.
• Ellipse has one main and minor axis as well as a centre.

Standard Equation of an Ellipse

We can easily calculate the ellipse equation when the ellipse's centre is at the origin (0,0) and the foci are on the x- and y-axes.

An ellipse's standard equation is expressed as;

x2/a2 + y2/b2 = 1