# Coordinate Geometry

Coordinate Geometry
Go back to  'Geometry'

Every place on this planet has coordinates that help us to locate it easily on the world map.

The coordinates of a few important places are as follows:

Boston - $$42.3601^ \circ N$$, $$71.0589 ^\circ W$$

Bengaluru - $$12.9716^ \circ N$$, $$77.5946 ^\circ E$$

Singapore - $$1.3521 ^\circ N$$, $$103.8198 ^\circ E$$

London -  $$51.5074 ^\circ N$$,  $$0.1278 ^\circ W$$

The coordinate system of our earth is made up of imaginary lines called latitudes and longitudes.

The zero degrees 'Greenwich Longitude' and the zero degrees 'Equator Latitude' are the starting lines of this coordinate system.

Here, in our study of the coordinate system in math we have the horizontal $$x$$-axis and the vertical $$y$$-axis as the reference lines.

## POINT LOCATOR - Simulation

Let us explore the simulation given below, to gain an initial understanding of the coordinate system.

With the help of your cursor, move the point on the graph and take note of the coordinates of the point on each of the axis, and also in the four quadrants.

In this mini-lesson, we shall explore the topic of coordinate geometry, by finding answers to questions like what is a coordinate plane, what is the difference between a scalar and a vector, and aim to understand the formulae used in coordinate geometry.

## What Is a Coordinate?

A coordinate is an address, which helps to locate a point in space.

For a two dimensional space, the coordinates of a point are $$(x, y)$$.

Here let us take note of these two important terms.

• Abscicca: It is the $$x$$ value in the point $$(x, y)$$.
• Ordinate:  It is the $$y$$ value in the point $$(x, y)$$.

The $$x$$ coordinate is called the abscissa and the $$y$$ coordinate is called the ordinate.

## What Is a Cartesian Plane Or Coordinate Plane?

A cartesian plane divides the space into two dimensions and is useful to locate the points.

It is also referred to as the coordinate plane

The features of the coordinate plane are:

• The plane consists of a horizontal axis called the $$x$$-axis, and a vertical axis called the $$y$$-axis.
• The point of intersection of $$x$$-axis and $$y$$-axis is the origin.
• The coordinates of the origin are $$(0, 0)$$.
• The axis divides the plane into four quadrants

## What Is the Difference Between Scalars and Vectors?

Scalars are simple numeric quantities that are represented on a unidimensional number line.

Vectors are the quantities with both numeric value and direction.

Vectors are generally represented as $$a \hat i + b \hat j$$.

Here $$a$$ and $$b$$ are the number values and $$\hat i$$ and $$\hat j$$ are the unit vectors representing the directions along the $$x$$ axis and $$y$$ axis respectively.

### Examples

• The marks of a student, the currency value, or the volume of a liquid can all be referred to as scalar quantities.
• The variable x = 3 is a scalar quantity.
• The speed of an aircraft has both a numeric value and also the direction associated with it and hence it can be called a vector.
• Also,  $$\bar A = 3 \hat i + 4 \hat j + 5 \hat k$$ represents a vector in maths.

## Equation and Slope of a Line

The slope of a line inclined at an angle $$\theta$$ with the positive $$x$$-axis is:

 $$m = \tan\theta$$

The slope of a line passing through the two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is:

 $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

The equation of a line passing through the point$$(x_1, y_1)$$ and having a slope m is:

 $(y - y_1) = m(x - x_1)$

The equation of a line passing through the points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is:

 $(y - y_1) = \frac{(y_2 - y_1)}{(x_2 - x_1)}.(x - x_1)$

The equation of a line cutting the $$y$$-axis at the point $$(0, c)$$ and having a slope $$m$$ is:

 $y = mx + c$

This equation is referred to as the slope-intercept form of equation of a line.

## Distance Formula

The distance between two points $$(x_1, y_1)$$ and $$x_2, y_2)$$ is as follows:

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

### Example

Find the distance between the points $$(-2, 2)$$ and $$(7, -4)$$.

Solution:

The coordinates of the given points can be represented as follows:

$$(x_1, y_1)$$ = (-2, 2) and $$(x_2, y_2)$$ = (7, -4)

Using the distance formula we can find the distance between these points.

\begin{align}d &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 }\\ &= \sqrt {(7 - (-2))^2 + (-4 - 2)^2 }\\ &=\sqrt {9^2 +(- 6)^2} \\ &= \sqrt{81 + 36} \\ &= \sqrt {117 }\end{align}

Therefore the distance between the two points is $$\sqrt {117}$$ units.

## Section Formula

The section formulae is useful to find the coordinates of a point which divided the join of the points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ in the ratio $$m : n$$

 $(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$

### Example

Find the point which divides the line joining the points $$(-5, 4)$$ and $$(2, 3)$$ in the ratio $$2 : 3$$

Solution:

The coordinates of the given points can be represented as follows.

$$(x_1, y_1) = (-5, 4)$$, $$(x_2, y_2) = (2, 3)$$ and $$m : n = 2 : 3$$

Here we apply the section formula for the given points.

\begin{align} (x, y) &= \left (\dfrac{mx_2 + nx_1}{m + n}, \dfrac{my_2 + ny_1}{m + n}\right) \\ &= \left(\dfrac{2(2) + 3(-5)}{2 + 3}, \dfrac{2(3) + 3(4)}{2 + 3}\right) \\ &=\left(\dfrac{4 -15}{5}, \dfrac{6 + 12}{5}\right) \\ &=\left(\dfrac{-11}{5}, \dfrac{18}{5}\right)\end{align}

Therefore, the point which divides the line joining the two points in the given ratio is $$\left(\dfrac{-11}{5}, \dfrac{18}{5}\right)$$.

## Midpoint Formula

The formula to find the midpoint of the line joining the points  $$(x_1, y_1)$$ and $$x_2, y_2)$$ is as follows.

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

### Example

Find the midpoint of the line joining the points $$(-2, 7)$$ and $$(8, 3)$$

Solution:  Applying the notation for the given points we have $$(x_1, y_1) = (-2, 7)$$, and $$(x_2, y_2) = (8, 3)$$

Here we can apply the midpoint formula.

\begin{align}(x, y) &=\left (\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\\&=\left(\dfrac{-2 + 8}{2}, \dfrac{7 + 3}{2}\right)\\&=\left(\dfrac{6}{2}, \dfrac{10}{2}\right)\\&=(3, 4)\end{align}

Therefore the midpoint is $$(3, 4)$$

Important Notes
1. The slope of $$x$$-axis is $$0$$ and the slope of $$y$$-axis is $$\infty$$.
2. The equation of $$x$$-axis is $$y = 0$$ and the equation of $$y$$-axis is $$x = 0$$
3. A point on the $$x$$-axis is of the form $$(a, 0)$$, and a point on the $$y$$-axis is of the form $$(0, b)$$
4. Point Slope Form of equation of a line is  $$(y - y_1) = m(x - x_1)$$.
5. Two Point Form of equation of a line is $$y - y_1 = \left(\dfrac{y_2 - y_1}{x_2 - x_1}\right).(x - x_1)$$
6. Slope Intercept Form of equation of a line is $$y = mx + c$$

## Solved Examples

 Example 1

Ron is given the coordinates of one end of the diameter of a circle as (5, 6) and the center of the circle as (-2, 1).

How can we help Ron to find the other end of the diameter of the circle?

Solution

Let $$AB$$ be the diameter of the circle with the coordinates of points $$A$$, and $$B$$ as follows.

$$A = (x_1, y_1)$$, $$B = (x_2, y_2) = (5, 6)$$

The coordinates of the center $$O = (x, y) = (-2, 1)$$

The formula for midpoint of the line is:

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

Applying this we have the following calculations.

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

Here we shall segregate the coordinates and the $$x$$ value is:

\begin{align} \dfrac{x_1 + 5}{2} &= -2 \\x_1 + 5 &= -2 \times 2\\x_1 + 5 &=-4 \\ x_1 &=-4 -5 \\x_1 &= -9 \end{align}

And the $$y$$ value is:

\begin{align} \dfrac{y_1 + 6}{2} &= 1 \\y_1 + 6&= 1 \times 2\\y_1 + 6 &=2 \\ y_1 &=2 - 6 \\y_1 &= -4 \end{align}

Therefore the point $$A = (x_1, y_1) = (-9, -4)$$

 $$\therefore$$ The other end of the diameter is $$(-9, -4)$$
 Example 2

Find the equation of a line passing through (-2, 3) and having a slope of -1.

Solution

The point on the line is $$(x_1, y_1) = (-2, 3)$$, and the slope is $$m = -1$$.

Applying the point and slope form of the equation of the line, we have:

\begin{align}(y - y_1) &= m(x - x_1) \\ (y - 3) &=(-1)(x -(-2)) \\ y - 3 &= -(x + 2) \\ y - 3 &= -x -2 \\ x + y &= 3 - 2 \\ x + y &= 1\end{align}

 $$\therefore$$ The equation of the line is $$x + y = 1$$
 Example 3

Find the equation of a line having a slope of -2 and $$y$$-intercept of 1.

Solution

The given information is $$m = -2$$ and $$y$$-intercept is $$c = 1$$

\begin{align} y &= mx + c \\ y &= (-2)x + 1 \\ y &= -2x + 1 \\ 2x + y &= 1\end{align}

 $$\therefore$$ The equation of the line is $$2x + y = 1$$

Challenging Questions
1. Find the centroid of a triangle having the coordinates of the vertices as A$$(2, 3)$$, B $$(4, -5)$$ and C$$(9, 11)$$
2. Find the$$x$$-intercept and the $$y$$-interceptor of the line having the equation $$6x + 5y-30 = 0$$

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of coordinate geometry. The math journey around coordinate geometry starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is a cartesian plane?

A cartesian plane consists of a coordinate axis with the horizontal $$x$$-axis and the vertical $$y$$-axis.

The point of intersection of these axes is the origin and all the points in the plane is represented with reference to the origin.

## 2. What is coordinate geometry?

Coordinate geometry is a branch of mathematics that helps to represent the points, lines, geometric figures in the coordinate system and to study their geometric properties.

## 3. What is abscissa?

The $$x$$-coordinate of a point $$(x, y)$$ in coordinate geometry is called abscissa. Numerically, it is the distance of the point on the $$x$$-axis from the origin.

## 4. What is ordinate?

The $$y$$-coordinate of a point $$(x, y)$$ in coordinate geometry is called ordinate. Numerically, it is the distance of the point on the $$y$$-axis from the origin.

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