Analytic Geometry is a branch of algebra, which deals with the modelling of some geometrical objects, such as lines, points, curves, and so on. It is a mathematical subject that uses algebraic symbolism and methods to solve the problems. It establishes the correspondence between the algebraic equations and the geometric curves. It covers some important topics such as midpoints and distance, parallel and perpendicular lines on the coordinate plane, dividing line segments, distance between the line and a point, and so on. In this article, let us discuss the terms used in the analytic geometry, formulas, cartesian plane, applications, and some solved problems.
Table of Contents:
- Definition
- Planes
- Coordinates
- Cartesian Plane
- Formulas
- Analytic Geometry in 3D
- Applications
- Problems
What is Analytic Geometry?
Analytic geometry is that branch of Algebra in which the position of the point on the plane can be located using an ordered pair of numbers called as Coordinates. This is also called coordinate geometry or cartesian geometry. This geometry is a contradiction to the synthetic geometry, where there is no use of coordinates or formulas. It is considered axiom or assumptions, to solve the problems.
Coordinate geometry has its use in both two dimensional and three-dimensional geometry. It is used to represent geometrical shapes. Let us learn the terminologies used in analytic geometry, such as;
- Plane
- Coordinates
Planes in Analytic Geometry
To understand how analytic geometry is important and useful, First, We need to learn what a plane is? If a flat surface goes on infinitely in both the directions, it is called a Plane. So, if you find any point on this plane, it is easy to locate it using Analytic Geometry. You just need to know the coordinates of the point in X and Y plane.
Coordinates in Analytic Geometry
Coordinates are the two ordered pair, which defines the location of any given point in a plane. Let’s understand it with the help of the box below.
A | B | C | |
1 | |||
2 | x | ||
3 |
In the above grid, The columns are labelled as A, B, C, and the rows are labelled as 1, 2, 3.
The location of letter x is B2 i.e. Column B and row 2. So, B and 2 are the coordinates of this box, x.
As there are several boxes in every column and rows, but only one box has the point x, and we can find its location by locating the intersection of row and column of that box.
- Cartesian Coordinates
- Polar Coordinates
- Cylindrical Coordinates
- Spherical Coordinates
Cartesian Coordinates
The most well-known coordinate system is the Cartesian coordinate to use, where every point has an x-coordinate and y-coordinate expressing its horizontal position, and vertical position respectively. They are usually addressed as an ordered pair and denoted as (x, y). We can also use this system for three-dimensional geometry, where every point is represented by an ordered triple of coordinates (x, y, z) in Euclidean space.
Polar Coordinates
In the case of polar coordinates, each point in a plane is denoted by the distance ‘r’ from the origin and the angle θ from the polar axis.
Cylindrical Coordinates
In the case of cylindrical coordinates, all the points are represented by their height, radius from z-axis and the angle projected on the xy-plane with respect to the horizontal axis. The height, radius and the angle are denoted by h, r and θ, respectively.
Spherical Coordinates
In spherical coordinates, the point in space is denoted by its distance from the origin ( ρ), the angle projected on the xy-plane with respect to the horizontal axis (θ), and another angle with respect to the z-axis (φ).
Cartesian Plane
In coordinate geometry, every point is said to be located on the coordinate plane or cartesian plane only.
Look at the figure below.
The above graph has x-axis and y-axis as it’s Scale. The x-axis is running across the plane and Y-axis is running at the right angle to the x-axis. It is similar to the box explained above.
Let’s learn more about Co-ordinates:
Origin: It is the point of intersection of the axis(x-axis and y-axis). Both x and y-axis are zero at this point.
Values of the different sides of the axis: x-axis – The values at the right-hand side of this axis are positive and those on the left-hand side are negative. y-axis – The values above the origin are positive and below the origin are negative.
To locate a point: We need two numbers to locate a plane in the order of writing the location of X-axis first and Y-axis next. Both will tell the single and unique position on the plane. You need to compulsorily follow the order of the points on the plane i.e The x coordinate is always the first one from the pair. (x,y).
If you look at the figure above, point A has a value 3 on the x-axis and value 2 on the Y-axis. These are the rectangular coordinates of Point A represented as (3,2).
Read: Analytic Function
Analytic Geometry Formulas
Graphs and coordinates are used to find measurements of geometric figures. There are many important formulas in coordinate Geometry. Few of the important ones are being used to find Distance, Slope or to find the equation of the line.
Distance Formula
Let the two points be A and B, having coordinates to be (x_{1},y_{1}) and (x_{2},y_{2}) respectively.
Thus, the distance between two points is given as-
d = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]
Midpoint Theorem Formula
Let A and B are some points in a plane, which is joined to form a line, having coordinates (x_{1},y_{1}) and (x_{2},y_{2}), respectively. Suppose, M(x,y) is the midpoint of the line connecting the point A and B then its formul is given by;
M(x,y) = [(x_{1}+x_{2}/2),(y_{1}+y_{2}/2)]
Angle Formula
Let two lines have slope m_{1} and m_{2} and θ is the angle formed between the two lines A and B, which is represented as;
tan θ = m_{1}-m_{2}/1+m_{1}m_{2}
Section Formula
Let two lines A and B have coordinates (x_{1},y_{1}) and (x_{2},y_{2}), respectively. A point P the two lines in the ratio of m:n, then the coordinates of P is given by;
Analytic Geometry of Three Dimensions
In this, we consider triples (a,b,c) which are real numbers and call this set as three- dimensional number space and denote it by R’. All the elements in the triple are called coordinates.
Let’s see how three-dimensional number space is represented on a geometric space.
In three-dimensional space, we consider three mutually perpendicular lines intersecting in a point O. these lines are designated coordinate axes, starting from 0, and identical number scales are set up on each of them.
Learn more on Coordinate Geometry in Two Dimensional Plane
Analytic Geometry Applications
Analytical geometry has made many things possible like the following:
- We can find whether the given lines are perpendicular or parallel.
- We can determine the mid-point, equation, and slope of the line segment.
- We can find the distance between the points.
- We can also determine the perimeter or the area of the polygon formed by the points on the plane.
- Define the equations of ellipse, curves, and circles.
Analytic Geometry Problems
Example 1:
What is the point of intersection of the axis (X-axis and Y-axis) called?
Solution:
The point of intersection of the axis (X-axis and Y-axis) called Origin and X and the Y-axis is 0 at this point.
Example 2:
Find the distance between two points A and B such that the coordinates of A and B are (5, -3) and (2, 1).
Solution:
Given that, the coordinates are:
A = (5, -3) = (x_{1}, y_{1})
B = (2, 1) = (x_{2},y_{2})
The formula to find the distance between two points is given as:
Distance,d = √[(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}]
d = √[(2-5)^{2}+(1- (-3))^{2}]
d =√[(-3)^{2}+(4)^{2}]
d =√[9+16]
d =√(25)
d = 5
Thus, the distance betweem two points A and B is 5.
Example 3:
Determine the slope of the line, that passes through the point A(5, -3), and it meets y-axis at 7.
Solution:
Given that, the point is A = (5, -3)
We know that, if the line intercepts at y-axis, then x_{2} = 0
Thus, (x_{2},y_{2}) = (0, 7)
The formula to find the slope of a line is:
m = (y_{2 }–y_{1}_{)/ }(x_{2 }-x_{1})
Now, substitute the values
m = (7- (-3))/(0_{ }– 5)
m = 10/-5
m = -2
Therefore, the slope of the line is -2.
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