Analytical Geometry—Two Dimensions

This easy-to-understand article aims to help students understand and grasp the concepts of analytical geometry, conic sections in 2D analytical geometry, and two dimensions—eccentricity and axis of a conic.

The eccentricity (denoted by ‘e’) of a conic section can be defined as the ratio of the distance from any point, (let’s call it point P) on the conic to its focus and the distance from that point P to the nearest directrix of a conic (i.e. the length of the perpendicular drawn from point P to that directrix).

Any conic has axes that can be defined according to the relation of axes with its focus.

Introduction to the concept of analytical geometry.

• 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 geometric problem.

• 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.

Introduction to the concept of the two-dimensional coordinate system and conic sections.

2D Coordinate System

• In two-dimensional coordinate geometry, every point in two-dimensional space is assigned unique coordinates that are used to identify the point 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 known as integers.

Conic Sections in 2D Coordinate geometry.

• Conic sections refer to the study of curves in analytical geometry, often called conics, as these curves can be found by intersections of a plane with a double-napped right circular cone.

• These curves can be circles, ellipses, parabolas, or hyperbolas.

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

• A conic can be defined as the locus of points for which the ratio of the distance to the focus (fixed point) and the distance to the directrix is a constant value. This ratio is known as the eccentricity of a conic.

The eccentricity of a conic.

• Properties of eccentricity:

  • The value of eccentricity remains constant for any conic

  • Eccentricity is a non-negative, real number for a conic

  • Eccentricity can define the shape of a given conic as well, i.e.

1. If e = 1, the given conic is a parabola

2. If e < 1, the given conic is an ellipse

3. If e > 1, the given conic is a hyperbola

4. If e = 0, the given conic is a circle

5. For a pair of lines, e = ∞

• The directrix of a conic can be defined as the fixed-line to which a perpendicular is drawn from point P.

• The distance from the focus to the corresponding directrix is known as the focal parameter and is denoted by ‘p’.

• If a cone is rotated along its vertical axis, the formula for eccentricity can be expressed as,

e = (sin β / sin α); 0 < α < π/2, 0 < β < π/2.

Here, β is the angle of the intersecting plane with the horizontal, whereas α is the angle between the cone’s slant generator and the horizontal.

• The linear eccentricity, denoted by ‘c’, is the distance between a conic’s centre and any one of its foci. The relation between eccentricity and linear eccentricity can be expressed as,

e = (c/a); where a is the length of the semimajor axis.

Axis of a conic

• Any conic has axes that can be defined according to the relation of axes with its focus.

• The axes are:

  • Principal axis: The principal axis is the line joining the two foci of a conic. The midpoint of a principal axis is the centre of the curve.

  • Major axis: It is the chord connecting the two vertices of a conic. Its length is equal to ‘2a’ units.

  • Semi-major axis: It is half the length of a major axis and is equal to ‘a’ units.

  • Minor axis: It is the length of the shortest diameter of a conic (generally, an ellipse), and its length is ‘2b’ units.

  • Semi minor axis: Half the length of a minor axis equals ‘b’ units.

Relationships between the two dimensions—eccentricity and axis of a conic.

1. l = pe; where l is the latus rectum

2. c = ae

3. (p + c) = a/e

4. e = (a/d); where d is the distance between the centre and the directrix

5. e = √ [f (2 – f)]; where f is flattening and is given by f = 1 – (b/a)

Conclusion

The eccentricity (denoted by ‘e’) of a conic section can be defined as the ratio of the distance from any point on the conic to its focus and the distance from that point to the nearest directrix of a conic.

Any conic has axes that can be defined according to the relation of axes with its focus. In a 2D conic, these axes are the principal axis, major axis, semi-major axis, minor axis, and semi-minor axis.

We are now familiar with terms like eccentricity, principal axis, semi-major axis, semi-minor axis and directrix and have also learnt the various relationships between two dimensions—eccentricity and axis of a conic.