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Analytical Geometry

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Analytical geometry or analytic geometry is also known as Cartesian geometry or coordinate geometry. It is a branch of mathematics which is a link between algebra and geometry. Analytical geometry deals with algebraic representation of geometrical figures. It pertains to representation of geometrical figures on a coordinate system. It deals with coordinates, positions of geometrical figures, algebraic equations and formulas, and various other applications related to fusion of algebra and geometry. In simple words, analytic geometry is analytical study of geometrical figures with respect to coordinate system.

Coordinate Systems

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Coordinate system is a system which specifies positions of geometrical figures in n-dimensional space using coordinates. Following three are the most commonly used coordinate systems:
  • Cartesian coordinate System: Cartesian coordinate system specifies location of a geometrical figure by using coordinates that are defined among mutually perpendicular lines. This system can be two dimensional or three dimensional. The coordinates in two-dimensional Cartesian coordinate system are denoted by (x, y) and are referred as ordered pair, while in three-dimensional Cartesian coordinate system, coordinates are represented by (x, y, z).
  • Polar Coordinate System: Polar coordinate system is a two-dimensional space where coordinates of a point are represented as distance from a fixed point and angle subtended at a fixed direction. Coordinates in this system are denoted by (r, $\theta$).
  • Spherical Coordinate System: Spherical coordinate system is a three-dimensional coordinate system where coordinates of a point are expressed as distance from a fixed point, angle formed at a fixed direction and angle of projection line of the point on third plane which is called azimuthal angle. Coordinates in this system are written as (r, $\theta$, $\phi$).
  • Cylindrical Coordinate System: Cylindrical coordinate system is also a three-dimensional space where coordinates are expressed as (r, $\theta$, z) or ($\rho$, $\phi$, z) where r or $\rho$ is radial coordinate, while $\theta$ or $\phi$ is azimuthal coordinate.

Distance Formula

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In analytical geometry, distance formula measures the distance between two points whose endpoints coordinates are specified as (x1, y1) and (x2, y2).
Distance Formula Image
The distance formula is given below:
Distance Formula
d is the distance between two points.

Let us consider one example.

Solved Example

Question: Find the distance between the points (1, -2) and (3, 1).
Solution:
 
Here, x1 = 1, y1 = -2 and x2 = 3, y2 = 1
Distance formula:
d = $\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$
d = $\sqrt{(3-1)^{2}+(1-(-2))^{2}}$
d = $\sqrt{(2)^{2}+(3)^{2}}$
d = $\sqrt{4+9}$
d = $\sqrt{13}$ = 3.61 units