Essential Formulas and Problem-Solving Strategies Explained
Coordinate Geometry is the branch of mathematics that helps us to exactly locate a given point with the help of an ordered pair of numbers. 2-dimensional geometry is the combination of geometry and algebra to solve the problems.
Two-dimensional Coordinate geometry is also said to be the study of graphs. Graphs are visual representations of our data. It can be in different forms like bar graphs, histograms, line graphs, etc.Two-dimensional coordinate geometry involves assigning each and every point in two-dimensional space with coordinates that can be uniquely used to identify the point. In relation to Coordinate axes (two perpendicular reference lines )coordinates are measured.
Equations of Coordinate Geometry
Given below are some important equations of coordinate geometry:
The equation of a line is ax + by + c = 0
The equation of a circle is x² + y² = r², where r is the radius of a circle.
The equation of a parabola is y²=4ax.
The equation of an Ellipse is x²/a²+y²/b²=1.
The equation of a Hyperbola is x²/a²-y²/b²=1.
Circle: A set of all points in a coordinate plane equidistant from a fixed point in the plane is defined as a Circle.
Parabola: A set of fixed points plane and all the points in the coordinate plane that are equidistant from the fixed straight line is defined as a Parabola.
Ellipse: A set of all points in a plane where the sum of the distances between two fixed points in the plane is constant is defined as an Ellipse.
Hyperbola: A set of all points in a plane where the difference in distance between two fixed points in the plane is constant is known as a Hyperbola.
Two-dimensional Coordinate geometry is also known as cartesian geometry.
Cartesian Coordinates (Rectangular Coordinates)
Cartesian Coordinates can be defined as a representation of a point in a plane by an ordered pair of real numbers.
Consider the below figure,
(Image will be uploaded soon)
Some of the Important Two-Dimensional Coordinate Geometry Terms
From this figure, let us understand some important terms used in 2-dimensional geometry.
Axes of Coordinates
In the above figure OX and OY are called X-axis and Y-axis respectively. And both together are known as axes of coordinates.
Origin
The point of intersection of the axes is called the origin; it is O.
Abscissa
The distance of any point on the plane from the Y-axis is called the abscissa.
Ordinate
The distance of any point on the plane from the X-axis is said to be ordinate.
Coordinate of the Origin
It has zero distance from both the axes. Hence the coordinates of the origin are (0, 0).
Quadrant
The axes divide the plane into four parts. These four parts are said to be quadrants.
A quadrant is ¼ th the part of a plane divided by coordinate axes.
Distance Between Two Points
Let two points be A (x1,y1) and B( x2,y2).
Then Distance Formula is Given by
d = \[ \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{^{2}}}\]
Section Formula
Let A(x1 , y1) and B(x2, y2) be the two pints on the XY plane. And P( x, y) be any point on segment AB such that AP: PB = m:n.
Condition 1: If the point P divides segment AB internally in the ratio m:n.
Then the (x,y) coordinates of P are given by:
\[\left [ \frac{mx_{2}+nx_{1}}{m+n},\frac{my_{2}+my_{1}}{{m+n}} \right ]\]
Condition 2: If the point P divided the line segment AB externally in the ratio m:n.
Then the ( x, y) coordinates of P are given by
\[\left [ \frac{mx_{2}-nx_{1}}{m-n},\frac{my_{2}-my_{1}}{{m-n}} \right ]\]
Condition 3: If the point P divided the line segment AB in the equal ratio i.e m = n or m:n = 1:1 then the ( x, y) coordinates of P are given by
P(x,y)=\[\left (\frac{x_{1}+x_{2}}{2} , \frac{y_{1}+y_{2}}{2}\right )\]
Area of a Triangle
The area of a triangle whose coordinates of vertices are (x1, y1) , (x2, y2) and (x3, y3)
The area of a triangle ABC whose vertices are A(x1, y1) , B(x2, y2) and C(x3, y3) is given by
Area of \[\Delta\] =\[\frac{1}{2}\left [ x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}) \right ]\]
Note :
To find the area of a polygon we divide it in triangles and take the numerical value of the area of each of the triangles.
Three points A(x1, y1) , B(x2, y2) and C(x3, y3) are collinear if and only if
Area of \[\Delta\] = 0 i.e \[\frac{1}{2}\left [ x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}) \right ]\] = 0
Solved Examples
Example 1 : P (4, 5) and Q(7, – 1) are two given points and the point R divides the line-segment PQ externally in the ratio 4:3. Find the coordinates of Y.
Solution :
Given that P(4,5)=(x1,y1),
Q(7,-1)=(x2,y2)
Point R divides the segment PQ in the ratio 4:3,
hence m=4, n=3
Since given that the point R divides the segment externally we use the section formula for external division,
R= \[\left \{ \left [ \frac{mx_{2-nx_{1}}}{m-n} \right ],\left [ \frac{(my_{2-ny_{1}})}{(m-n)}\right ] \right \}\]
Substituting the given values,
= \[\left \{\frac{4\times 7-3\times 4}{4-3} , \frac{4\times \left ( -1 \right )-3\times 5}{4-3}\right \}\]
= \[{\frac{(28-12)}{1},\frac{(-4-15)}{1}}\]
= {16,-19}
The coordinates for the point R which divides the segment PQ externally are (16,-19)
Example 2 : Find the area of the LMN whose vertices are L(3, 2), M(4, 2) and N(3, 5)
Solution:
Given that L( 3, 2) = ( x1, y1)
M( 4, 2) = ( x2, y2)
N( 3, 5) = ( x3, y3)
To calculate area of LMN we have formula
To calculate area of LMN we have formula
A = \[\frac{1}{2}\left [ x_{1}(y_{2}-y_{3})+x_{2}(y_{3}-y_{1})+x_{3}(y_{1}-y_{2}) \right ]\] = 0
A = \[ \frac{1}{2} 3(2-5)+4(5-2)+3(2-2)\]
A = \[\frac{1}{2}\left [ 3.(-3)+4.3+1 \right ]\]
A= \[\frac{1}{2}\left [ -9+12 \right ]\]
A = \[\frac{1}{2}\left [ 3 \right ]\]
A=\[ \frac{3}{2}\]
Therefore, the area of a triangle LMN is 3/2 square units.
Conclusion:
This article talks about Two Dimensional Coordinate Geometry, their equations, and examples. You can refer to this for furthering your understanding of the topic.
FAQs on Two Dimensional Coordinate Geometry Made Easy
1. What is a two-dimensional (2D) coordinate system?
A two-dimensional coordinate system, also known as the Cartesian plane, is a framework used to uniquely define the position of any point on a flat surface. It is formed by the intersection of two perpendicular number lines: a horizontal line called the x-axis and a vertical line called the y-axis. Their point of intersection is called the origin (0,0).
2. What are the key components that define a point's location in 2D coordinate geometry?
The location of a point is defined by its coordinates, which are an ordered pair of numbers (x, y). The key components are:
Abscissa (x-coordinate): The perpendicular distance of the point from the y-axis. It indicates the horizontal position.
Ordinate (y-coordinate): The perpendicular distance of the point from the x-axis. It indicates the vertical position.
Quadrants: The four regions into which the axes divide the plane, each with a specific sign convention for the coordinates.
3. Why are the coordinates in 2D geometry written specifically as an 'ordered pair' (x, y)?
The term 'ordered pair' is critical because the sequence of the numbers matters fundamentally. The first number always represents the horizontal displacement (x-value), and the second number always represents the vertical displacement (y-value). Interchanging them changes the location entirely. For example, the point (2, 5) is in a completely different position from (5, 2). This strict order ensures every point on the plane has a single, unambiguous address.
4. What are some of the fundamental formulas used in two-dimensional coordinate geometry?
Several fundamental formulas allow us to analyse the relationships between points and lines on the Cartesian plane. The most important ones include:
Distance Formula: To calculate the distance between two points (x₁, y₁) and (x₂, y₂), we use: D = √[(x₂ - x₁)² + (y₂ - y₁)²].
Midpoint Formula: To find the coordinates of the midpoint of a line segment joining (x₁, y₁) and (x₂, y₂), we use: [(x₁ + x₂)/2, (y₁ + y₂)/2].
Section Formula: To find a point that divides a line segment in a given ratio m:n, we use: [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)].
Area of a Triangle: To find the area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃).
5. How does coordinate geometry serve as a bridge between algebra and geometry?
Coordinate geometry creates a powerful bridge between the abstract world of algebra and the visual world of geometry. It achieves this by representing geometric shapes as algebraic equations. For instance, a straight line can be expressed as a linear equation (e.g., y = 2x + 3), and a circle can be described by an equation like (x - h)² + (y - k)² = r². This allows us to use algebraic tools to solve geometric problems and prove geometric properties, transforming spatial relationships into numerical calculations.
6. What is the main difference between a 2D shape and its representation in coordinate geometry?
A 2D shape (like a square or circle) is a conceptual geometric figure with properties like side length and angles. Its representation in coordinate geometry is more concrete: it is a collection of points (vertices or the curve itself) whose coordinates all satisfy a specific algebraic equation or a set of conditions. For example, a square is a shape, but a square with vertices at (0,0), (4,0), (4,4), and (0,4) is its specific instance on the Cartesian plane.
7. Beyond textbook problems, what are some real-world examples of the importance of 2D coordinate geometry?
Two-dimensional coordinate geometry is foundational to many modern technologies and fields. Key examples include:
Mapping and GPS: It is the basis for digital maps like Google Maps, where locations are plotted as coordinates (latitude and longitude).
Computer Graphics and Gaming: Every pixel on your screen has a coordinate. Game developers use a 2D coordinate system to position characters, objects, and environments.
Engineering and Design (CAD): Architects and engineers use Computer-Aided Design (CAD) software, which relies on a coordinate system to create precise blueprints for buildings, machinery, and circuits.
