Coordinate geometry or Cartesian geometry is a branch of mathematics that also serves as an intermediary between geometry and algebra since points, lines, and geometrical shapes are positioned in the coordinate plane. Coordinate geometry aids in identifying the exact location of geometrical objects through the application of coordinates in ordered pairs (x, y). The y-axis (vertical) and the x-axis (horizontal) split the coordinate plane and cut at point (0,0), splitting it further into four quadrants. Coordinate geometry is applied to find many geometric properties such as distance, slope, midpoint, and area algebraically. The topic is very significant in physics, engineering, computer graphics, and even navigation systems.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with geometrical figures in a two-dimensional plane. This helps in learning about the properties of these figures.

Coordinate Plane

A cartesian plane divides the plane in two dimensions that allows locating of the points easily. The cartesian plane or coordinate plane works in two axes: a horizontal axis and a vertical axis, known as x-axis and y-axis. These axes of the coordinate plane divide the plane into four quadrants which intersect at a common point called origin with coordinates as (0,0).

Also, any other point on the coordinate plane is represented as (x,y), where the value of x is the position of the point with respect to x-axis, and the value of y is the position of the point with respect to y-axis.

Some of the properties of the points represented in the coordinate plane are:

• The point of intersection of the two axis is represented by the origin O with coordinates (0,0).
• Towards the right of the origin is the positive x-axis and on its opposite side is the negative x-axis.
• The y-axis above the origin is the positive y-axis and below origin is negative y-axis.
• The point in the first quadrant has both the coordinates positive and the points are represented by (x,y).
• The point in the second quadrant has x-coordinate negative and y-coordinate positive and the points are represented by (-x,y).
• The point in the third quadrant has both x, and y-coordinate negative and the points are represented by (-x,-y).
• The point in the fourth quadrant has x-coordinate positive and y-coordinate negative and the points are represented by (x,-y).

Coordinates of a Point

To locate any point in space, coordinates of the point acts as an address of the point. (x,y) are the coordinates of a point. Some of important terms linked with coordinates are:

Abscissa: The value of the x-coordinate of a point on the coordinate plane is called its abscissa. It is also known as the distance of the point from the x-axis.

Ordinate: The y-coordinate of a point on the coordinate plane is called its ordinate. It is also known as the distance of the point from the y-axis.

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Coordinate Geometry vs Euclidean Geometry

Coordinate geometry is that branch of mathematics where algebra meets geometry. Generally, when we study coordinate geometry, we work in a two dimensional Real number space. Additionally, coordinate geometry can also be used studying three-dimensional space.

However, euclidean geometry primarily deals with points, lines, and circles, that means basic geometrical figures and their properties. We can use the concept of euclidean geometry to find the areas, parameters, and other related information for 2-dimensional objects.

Coordinate geometry considers points as ordered pairs that are represented as (x,y), lines can be represented by equations like ax+ by + c = 0, and circles as \(\left(x-a\right)^2+\left(y-b\right)^2=r^2\), where (a,b) are the coordinates of the center of the circle and r is the radius.

Learn about x axis and y axis

Coordinate Geometry Formulae

With the help of the formulae in coordinate geometry we can prove various properties of lines and other fingers in the cartesian plane. Some of the common formulas studied under coordinate geometry are the distance formula, section formula, midpoint formula, and slope formula.

Let us learn about some of the formula in coordinate geometry:

Coordinate Geometry Distance Formula

In order to find the distance between the two points \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\) on the cartesian plane is written as the square root of the sum of the square of the difference between x-coordinates and the y-coordinates of the given point.

This can be represented as:

\(D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

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Section Formula in Coordinate Geometry

Section formula in coordinate is used for finding the coordinates of the point on a line with endpoints \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\) in the cartesian plane that divides the line in the ratio m:n. This point can lie either between these two points or outside the line segment joining these points.

The section formula in coordinate geometry is written as:

\(\left(x,y\right)\ =\ \left(\frac{mx_2+mx_1}{m+n},\frac{my_2+my_1}{m+n}\right)\)

Midpoint Formula in Coordinate Geometry

Midpoint formula in coordinate geometry is a special case in the section formula where the point divides the line in the ratio 1:1. For a line joining the points \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\), midpoint of a given line is obtained by finding the average of x-values of the two given points and the average of y-values of the two given points.

We can denote it by:

\(\left(x,y\right)\ =\ \left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)\)

Slope Formula

The inclination of the line with respect to the axis is called the slope of the line. We generally calculate the slope by finding the angle made by the line with the x-axis, or it can also be found by considering any two points on the line.

For a line inclined at an angle \(\theta\) to the x-axis, the slope is represented by \(m=-\tan\theta\).

For a line made by joining the two points \(\left(x_1,y_1\right)\) and \(\left(x_2,y_2\right)\), slope is:

\(m=\frac{(y_2−y_1)}{\left(x_2-x_1\right)}\).

Centroid of a Triangle

We know that the centroid of a triangle is the point of intersection of all the three medians of a triangle. And median of a triangle is the line joining the vertex of the triangle to the midpoint of the opposite side.

For a triangle with the three vertices as \(A\left(x_1,y_1\right),\ B\left(x_2,y_2\right),\ C\left(x_3,y_3\right)\), centroid is represented by:

\(\left(x,y\right)\ =\ \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)\)

Learn about Parabola Ellipse and Hyperbola

Equation of Shapes in Coordinate Geometry

Using the points on a given regular shape, its equation can be determined in coordinate geometry. We can find the equations of line, circle, parabola, hyperbola, etc using some specific formulae mentioned below:

Equation of Line in Coordinate Geometry

Equation of a line represents the positions of all the points on the line. The standard equation of a line is given as ax + by + c = 0.

However, there is yet another method to find the equation of a line. This is called the slope-intercept method. The equation of a line in the slope intercept form is given as:

y = mx + c

Here, m is the slope of the line, and c is the y-intercept of the line.

Equation of a Circle in Coordinate Geometry

With the help of a simple equation of a circle we get precise information about the center of the circle and the radius of the circle.

For a circle with center \(\left(x_1,y_1\right)\), and radius ‘r’, the equation of the circle is given by:

\(\left(x-x_1\right)^2+\left(y-y_1\right)^2=r^2\)

Here, (x,y) is any arbitrary point on the circumference of the circle.

Equation of a Parabola, Hyperbola, and Ellipse

Let us discuss the equation of a parabola, hyperbola and ellipse. We know that Parabola in geometry is a symmetric U-shaped curve, with every point on the figure being at an equal distance from a fixed point known as focus of the parabola.

When the directrix of parabola is parallel to y-axis, the standard equation of parabola is given as:

\(y^2=4ax\)

And, if the directrix is parallel to x-axis a parabola is represented by

\(x^2=4ay\).

In case the parabola is in the negative quadrants, the equations become:

\(y^2=-4ax\), and \(x^2=-4ay\).

Now, let us check the equation for a hyperbola:

Hyperbola is an open curve with two branches that are a mirror image of each other. Also, we can define hyperbola as a locus of point moving in a plane in a way that the ratio of its distance from a fixed point that is focused to that of a fixed line that is directrix is a constant that is greater than 1.

We can write the equation of an hyperbola in the simplest form when the center of the hyperbola is at the origin, and the focus lies on either of the axis.

The standard equation of a hyperbola is:

\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), with \(b^2=a^2(e^2-1)\).

Here (x,y) is the coordinate of any point on the hyperbola, and a is the focus.

Ellipse on the other hand is a geometrical figure that is defined as a locus of point that has a ratio between the distance from a fixed point and the fixed line as ‘e’, where e is the eccentricity of the ellipse.

The general equation of an ellipse is given as:

\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

Where (-a,0) and (a,0) are the end vertices of the major axis, and (0,b) and (0,-b) are the end vertices of the minor axis.

Area of Polygons in Coordinate Geometry

A polygon is a closed geometric figure that is made by joining a finite number of straight lines. In order to find the area of a polygon we name the vertices of the figure in a sequence either in the clockwise or anti-clockwise direction.

For a polygon figure with ‘n’ vertices denoted by \(\left(x_1,y_1\right),\ \left(x_2,y_2\right),…,\left(x_n,y_n\right)\).

The area is given by the formula:\(\left|\frac{\left(x_1y_2-y_1x_2\right)+\left(x_2y_3-y_2x_3\right)+…+\left(x_ny_1-y_nx_1\right)}{2}\right|\) sq. units.

Area of a Triangle in Coordinate Geometry

Triangle is a special type of polygon with three sides. For a triangle ABC with vertex \(A\left(x_1,y_1\right),\ B\left(x_2,y_2\right),\ and\ C\left(x_{3,}y_3\right)\), the area of a triangle is represented by:

Area of triangle ABC= \(\frac{1}{2}|x_1(y_2−y_3)+x_2(y_3−y_1)+x_3(y_1−y_2)|\) sq. units.

Area of a Quadrilateral in Coordinate Geometry

Quadrilateral is a geometrical figure with four sides and 4 vertices. For a quadrilateral ABCD with vertices \(A\left(x_1,y_1\right),\ B\left(x_2,y_2\right),\ C\left(x_{3,}y_3\right),\ and\ D\left(x_4,y_4\right)\) the area is represented by:

Area of a quadrilateral ABCD= \(\frac{1}{2}\left\{\left(x_1-x_3\right)\left(y_2-y_4\right)-\left(x_2-x_4\right)\left(y_1-y_3\right)\right\}\) sq. units.

Applications of Coordinate Geometry

Some of the common applications of coordinate geometry are:

• Used for figuring out the distance between two objects.
• Coordinate geometry is used in computer monitors. Some complex curves, shapes and conics are better interpreted with algebraic equations that would otherwise be difficult to analyze using pure geometry.
• Geometry finds its use in the human digestive system as it involves organizing of tubes within a tube.
• Using coordinate geometry we can easily locate and get the precise location of a place in the actual world.
• Air traffic is regulated using coordinate geometry. A slight movement in the aircraft up, down, left or right leads to the change in the position of the aircraft in the coordinate axis.

Coordinate Geometry Solved Examples

Que 1: The center of a circle and one end of the diameter is given as (-2,1) and (5,6) respectively. Using the formulas of coordinate geometry find the other end of the diameter of the circle?

Solution 1: Let AB be the diameter of a circle.
Let the coordinates of A and B are \((x_1,y_1)\) and \((x_2,y_2)\) respectively.

But we know that B\((x_2,y_2)\) = (5,6).

Let O be the center of the circle with coordinates= (-2,1)

The midpoint formula for a line is given by:

\(\left(x,y\right)\ =\ \left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)\)

Substituting the values:

\((-2,1)=\left(\frac{x_1+5}{2},\frac{y_1+6}{2}\right)\)

Comparing left hand side and right hand side, we get:

\(-2=\frac{x_1+5}{2}\)

\(-4=x_1+5\)

\(-9=x_1\),

And,

\(1=\frac{y_1+6}{2}\)

\(2=y_1+6\)

\(-4=y_1\)

Therefore, the coordinates of point A are (-9,-4).

So, the other end point of the diameter is A (-9,-4)

Que 2: Find the equation of the line with slope -2 and y-intercept as 1.

Solution 2: Given that m = -2, and c = 1.

Using slope-intercept form of a line, equation of a line is;

y = mx + c

y = (-2)x + 1

2x + y = 1.

So, the required equation of a line is 2x + y = 1.

Coordinate geometry connects algebra and geometry, allowing geometric shapes to be analyzed using algebraic equations. With the help of points, lines, and figures on a Cartesian plane, we are able to find distances, midpoints, slopes, and so on. Equations of circles, parabolas, hyperbolas, and ellipses give us more information about these figures. With its uses in GPS, computer graphics, and air traffic control, coordinate geometry increases problem-solving abilities and propels technological progress. We hope this article helps your understanding and preparation for the exam. Keep yourself updated with the Testbook App for information and test series access to gauge your knowledge.