JEE Important Chapter - Coordinate Geometry

The study of geometric figures using coordinate axes is known as coordinate geometry. Straight lines, curves, circles, ellipses, hyperbolas, and polygons can all be conveniently drawn and represented to scale in the coordinate axes. Furthermore, using the coordinate system to work algebraically and examine the characteristics of geometric figures is referred to as coordinate geometry.

In this article, We will go through some of the important concepts of co-ordinate geometry like what is coordinate geometry - definitions, explanations and formulas. Also, we will go through some of the solved coordinate geometry problems for a better understanding of the concept. 

JEE Main Maths Chapter-wise Solutions 2022-23 

Important Topics of Coordinate Geometry

• Cartesian Coordinates

• Distance Formula

• Distance Between Two Points

• Slope

• Midpoint formula

• Equation of a line 

• Slope-Intercept Form of a Line

• Point Slope Form

• Euclidean Distance Formula

Important Concepts of Coordinate Geometry

What is Coordinate Geometry?

Coordinate geometry is a field of mathematics that aids in the presentation of geometric shapes on a two-dimensional plane and to study their properties. To get a basic understanding of Coordinate Geometry, we will learn about the coordinate plane and the coordinates of a point initially.

Coordinate Plane

A cartesian plane divides plane space into two dimensions, making it easier to locate points. The coordinate plane is another name for it. The horizontal X-axis and the vertical Y-axis are the two axes of the coordinate plane. The origin is the place where these coordinate axes connect, dividing the plane into four quadrants (0, 0). Furthermore, any point in the coordinate plane is represented by a point (x, y), where the x value represents the point's position relative to the X-axis and the y value represents the point's position relative to the Y-axis.

The point represented in the four quadrants of the coordinate plane has the following properties:

• The origin ‘O’ is the point of intersection of the X-axis and the Y-axis and has the coordinate points (0, 0) generally.

• The X-axis to the right of the origin ‘O’ is the positive X-axis and to the left of the origin, ‘O’ is the -(ve) X-axis. Also, the Y-axis above the origin ‘O’ is the +(ve) Y-axis, and below the origin ‘O’ is the -(ve) Y-axis.

• The point present in the first quadrant (x, y) has both +(ve) values and is plotted with reference to the +(ve) X-axis and the +(ve) Y-axis. 

• The point represented in the second quadrant is (-x, y) is plotted with reference to the -(ve) X-axis and +(ve) Y-axis.

• The point represented in the third quadrant (-x, -y) is plotted with reference to the -(ve) X-axis and -(ve) Y-axis.

• The point represented in the fourth quadrant (x, -y) is plotted with reference to the +(ve) X-axis and -(ve) Y-axis.

Coordinate Geometry Formulas

Coordinate geometry formulae make it easier to prove the various properties of lines and figures represented by coordinate axes. The distance formula, slope formula, midpoint formula, section formula, and line equation are included in the list of coordinate geometry formulas. In the below paragraph, we'll learn more about each of the formulas.

Coordinate Geometry Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is equal to the square root of the sum of the squares of the difference between the X-coordinates and the Y-coordinates of the two given points. The formula for the distance between two points is as given below:

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

Slope Formula

The inclination of a line is measured by its slope. The slope can be determined by selecting any two locations on the line and measuring the angle formed by the line with the positive X-axis. m = tanθ is the slope of a line that is inclined at an angle 'θ' with the positive X-axis.  The slope of a line joining the two points $(x_1, y_1)$ and $x_2, y_2)$ is equal to m = $\frac {(y_2 - y_1)}{(x_2 - x_1)}$.

m = tanθ

m = $(y_2 - y_1)$/$(x_2 - x_1)$

Mid-Point Formula

The formula for finding the midpoint of the line connecting the points $(x_1, y_1)$ and $x_2, y_2)$ is a new point, whose abscissa is the average of the x values of the two given points, and the ordinate is the average of the y values of the two given points. The midpoint is on the line that connects the two points and is right in the middle of them.

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

Section Formula in Coordinate Geometry

The section formula is useful to find the coordinates of a point that divides the line segment joining the points $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m : n$. The point that divides the provided two points is located on the line that connects them and can be found either between the two points or outside the line segment between them.

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

The Centroid of a Triangle

The centroid of a triangle is the point of intersection of medians of a triangle. (Median is a line that joins the vertex of a triangle to the mid-point of the opposite side). The centroid of a triangle having its vertices A$(x_1, y_1)$, B$(x_2, y_2)$, and C$(x_3, y_3)$ is obtained from the following formula.

$(x, y) = (\dfrac{x_1+ x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3})$

Area of a Triangle Coordinate Geometry Formula

The area of a triangle having the vertices A$(x_1, y_1)$, B$(x_2, y_2)$, and C$(x_3, y_3)$ is obtained from the following formula. This formula is used to find the area of a triangle for all types of triangles.

Area of a Triangle = $\dfrac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

How to Find Equation of a Line in Coordinate Geometry?

With the use of a simple linear equation, this line equation represents all of the points on the line, ax + by + c= 0 is the standard form of a line equation. There are several methods for determining a line's equation. The slope-intercept form of the equation of a line (y = mx + c) is another essential form of the equation of a line. The slope of the line is m, and the Y-intercept of the line is c. The equation of a line also includes other types of line equations, such as point-slope form, two-point form, intercept form, and normal form.

y = mx + c

List of Formulae

Solved Examples 

Question 1. 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 $m = -1$.

From the coordinate geometry point and slope form of the equation of the line, we get:

$(y - y_1) = m(x - x_1) \\ (y - 3) =(-1)(x -(-2)) \\ y - 3 = -(x + 2) \\ y - 3 = -x -2 \\ x + y  = 3 - 2 \\ x + y  = 1$

Therefore the equation of the line is: x + y = 1

Question 2. 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 $c = 1$ 

From coordinate geometry we can make use of the slope intercept form of the equation of a line.

$y = mx + c \\ y = (-2)x + 1 \\ y = -2x + 1  \\ 2x + y = 1$

Therefore the equation of the line we get is 2x + y = 1.

Solved Problem of Previous year Question Paper 

Question 1. The equations of two equal sides of an isosceles triangle are 7x − y + 3 = 0 and x + y − 3 = 0 and the third side passes through the point (1, 10). Then the equation of the third side is ___________.

Solution:

Any line passing through the point (1, 10) is given by y + 10 = m (x − 1)

Since it makes an equal angle say $\alpha$ with the given lines 7x − y + 3 = 0 and x + y − 3 = 0, therefore tan α = $[$m − 7$]$ / $[$1 + 7m$]$

= $[$m − (−1)$]$ / $[$1 + m (−1)$]$

⇒ m = $[$1/3$]$ or 3

So, the two possible equations of the third side are: 3x + y + 7 = 0 and x − 3y − 31 = 0.

Question 2. The locus of a point P, which divides the line joining (1, 0) and (2cosθ, 2sinθ) internally in the ratio 2:3 for all θ, is a ________.

Solution:

Let us consider the coordinates of the point P, that divides the line joining (1, 0) and (2 cosθ, 2sinθ) in the ratio 2:3 be (h, k). Then,

h = $[$4 cosθ + 3$]$/$[$5$]$ and k = $[$4 sinθ$]$/$[$5$]$

cosθ = $[$5h − 3$]$/$[$4$]$ and sinθ = $[$5k$]$/$[$4$]$

($[$5h − 3$]$/$[$4$]$)2 + ($[$5k$]$/$[$4$]$)2 = 1

(5h − 3)2 + (5k2) = 16

Therefore, locus of (h, k) is (5x − 3)2 + (5y)2 = 16, which is a circle.

Question 3. If the slope of a line passes through the point A (3, 2) is 3/4, then the points on the line which are 5 units away from A, are ___.

Solution:

The equation of line passes through the point (3, 2) and of slope 3/4 is 3x − 4y − 1 = 0

Let the point be (h, k) then,

3h−4k−1 = 0 …………..(i)     and

(h − 3)2 + (y − 2)2 = 52 (ii)

On solving the equations above, we get h = −1, 7 and k = −1, 5.

So, the points are (1, 1) and (7, 5).

Practise Questions 

Question 1. If the coordinates of vertices of a triangle are (0, 5), (1, 4) and (2, 5) then the coordinate of the circumcentre will be.

A.  (1, 5)

B.  $\left ( \dfrac{3}{2},\: \dfrac{9}{2} \right )$

C.  (1, 4)

D.  None of these

Question 2. The equation of the image of pair of rays y = |x| by the line x = 1 is

A.  |y| = x + 2

B.  |y| + 2 = x

C.  y = |x – 2|

D.  None of these

Answers: 2-A, 3-C

Conclusion 

Coordinate Geometry is recognized as one of the most fascinating mathematical concepts. Coordinate Geometry topics uses graphs with curves and lines to describe the relationship between geometry and algebra. Geometric aspects are provided in Algebra, allowing students to solve geometric problems. It is a type of geometry in which the coordinate points on a plane are expressed as an ordered pair of numbers. Here, the concepts of coordinate geometry are explained along with their formulas and solved coordinate geometry examples for better understanding. 

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