Analytical Geometry-Two Dimensions — Ellipse and Hyperbola

Coordinate Geometry, also known as Analytical Geometry, is the field of mathematics that uses algebraic symbols and methods to represent and solve an equation for a problem in geometry.

Analytical geometry establishes a correspondence between geometric curves and algebraic equations, making it possible to reformulate geometric problems in terms of equivalent algebraic problems and vice versa.

Among the analytical geometry topics, two dimensions of ellipse and hyperbola are important concepts. 

But let us first understand the concepts of the two-dimensional coordinate system and conic sections briefly.

Two-Dimensional Coordinate System and Conic Sections

Let us get a brief idea about the two-dimensional coordinate systems and conic sections.

Two-Dimensional Coordinate System

Two-dimensional coordinate geometry assigns each point in a two-dimensional space with unique coordinates to identify it in the plane or graph.

The X-axis represents the horizontal axis line, whereas the Y-axis represents the vertical axis line. The values represented on the X and Y axes are integers.

Conic Sections in 2-D Coordinate Geometry

The study about curves in analytical geometry is called conic sections, often called conics. It is so named as we can find these curves at the intersections of a plane with a double-napped right circular cone.

These curves can be circles, ellipses, parabolas, or hyperbolas. In this article, we will be focusing on two dimensions – ellipse and hyperbola.

The applications of conic sections include using its concepts in planetary motion, design of telescopes and antennas, construction of reflectors in flashlights and automobile headlights, and many more.

Two Dimensions – Ellipse and Hyperbola

When the plane cuts a nappe of the cone which is not its vertex:

• If α < β < 90o; the section is an ellipse.

• If 0 ≤ β < α; i.e., the plane cuts through both the nappes and the curves of intersection is a hyperbola. 

• If 0 ≤ β < α, the section is a pair of intersecting straight lines, and it is a degenerated case of the hyperbola.

Here, on rotating a line m around another line l, the angle α represents the angle at the point of intersection of the two lines, and β is the angle made by the intersecting plane with the vertical axis of the cone.

Ellipse

An ellipse consists of all points set in a plane, whose sum of distances from two fixed points in the plane remains constant. The fixed points are known as foci. There is a line segment that joins both foci of an ellipse. The mid-point of this particular line segment is known as the center of an ellipse.

The line segment passing through the foci of an ellipse is the major axis. The line segment passing through the center, perpendicular to the major axis, is the minor axis. The endpoints of the major axis of an ellipse are known as an ellipse’s vertices.

The notations for the different lengths of the ellipse are

• 2a denotes the major axis.

• 2b indicates the minor axis.

• 2c represents the distance between the foci.

The length of the semi-major axis is half of the major axis and equal to a, and that of the semi-minor axis is half of the minor axis and equal to b.

The standard equation of an ellipse with the centre at origin and foci on either x or y-axis is:

(x2/b2) + (y2/a2) = 1

OR

(x2/a2) = 1 – (y2/b2)

Such that, x2 ≤ a2, so – a ≤ x ≤ a

The latus rectum of an ellipse is a line segment perpendicular to the major axis through the foci their endpoints lie on the ellipse. The latus rectum’s length for the ellipse’s standard equation is 2b2/a.

Hyperbola

All points set in the plane, such that the difference of their distances from two fixed points in the plane is constant, form a hyperbola. The ‘difference’ means the distance to the farther point minus that to the closest point.

The fixed points are known as foci. The mid-point of the line segment that joins foci is the center.

The line passing through the foci of a hyperbola is the transverse axis. The conjugate axis is the line passing through the center of a hyperbola, perpendicular to the transverse axis. The points of intersection of the transverse axis with the conjugate axis are the vertices of a hyperbola.

The lengths of the hyperbola are:

• 2a = transverse axis

• 2b = conjugate axis

• 2c = distance between the foci

Here, 

b = √ (c2 – a2)

We can express the standard equation of a hyperbola with the centre at origin as:

(x2/a2) – (y2/b2) = 1, foci on x-axis

OR

(y2/a2) – (x2/b2) = 1, foci on y-axis

A hyperbola in which a = b is called an equilateral hyperbola. The length of the latus rectum of a hyperbola is 2b2/a, i.e., similar to an ellipse.

Conclusion

This article helps understand the concepts of conic sections and two dimensions – ellipse and hyperbola in analytical geometry.

We now know that an ellipse comprises all points set in a plane, whose sum of distances from two fixed points in the plane remains constant. We can define a hyperbola as all points set in a plane, the difference of whose distances from two fixed points in the plane is a constant.

We are now also familiar with the standard equations and related terms and notations in two dimensions – ellipse and hyperbola.