Class 10 Coordinate Geometry Exercise Notes for Exams

Class 10 Mathematics Exercise Notes for Chapter 7 – Coordinate Geometry describes the various core concepts of coordinate geometry. Coordinate Geometry helps the students to interpret and quantitatively present geometrical shapes. It helps to extract numerical information by taking out logical conclusions. We have Class 10 Mathematics Exercise Notes curated by Expert Teachers. These are helpful for students in solving various exercises of this chapter for homework or exam practice. Also, these concepts build a foundation for further study & competitive exams like NTSE, Olympiad exam, KVPY, and more. The students can get easy access to our exercise notes for different topics. By practicing with our explanatory notes, students can attain perfection in Coordinate geometry. Also, these notes come in handy for your exam preparation. So, let us begin with it!

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What is Coordinate Geometry?

The branch of mathematics that deals with cartesian coordinates is known as Coordinate Geometry. Coordinate geometry has evolved as an algebraic means for studying the geometry of figures. It simply helps to study geometry by utilizing algebra and understand algebra by utilizing concepts of geometry. This system is most commonly used to exploit the equations of several two-dimensional figures like circles, triangles, squares, and more.

What will Students get to learn?

Class 10 Coordinate Geometry is a very crucial chapter in mathematics. It will help the students to find:

• The distance between the two points when their coordinates are given,
• The area of the triangle formed by three given points,
• The coordinates of the point which divides a line segment that joins the given points in a given ratio. 

The students must gain a good understanding of this lesson as it forms the base for various other related topics like Constructions in Class 10 Mathematics. And this can be done if you know the detailed JEE Main 2022 Syllabus and JEE Advanced 2022 Syllabus.

Key Terminologies in Coordinate Geometry

Some of the crucial terms interconnected with coordinate geometry that the students must clearly understand. Those crucial terminologies are:

• Coordinates: A set of values that helps to ascertain and represent the exact position of a point in the coordinate plane are known as Coordinates.
• Coordinate Plane: A Coordinate Plane is a 2D plane. It is formed by the intersection of two perpendicular lines which are called the x-axis and y-axis.
• Abscissa: Abscissa or x-coordinate is the distance of any point from the y-axis.
• Ordinate: Ordinate or y-coordinate is the distance of a point from the x-axis.

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Coordinate Geometry: Explanatory Notes

Understanding Coordinate and A Coordinate Plane

The students must be aware of plotting graphs on a plane. We are given values or numbers in a table form for both linear and nonlinear equations. Here, the number line which is also called a Cartesian plane features four quadrants formed by two intersecting axes which are perpendicular to each other. The horizontal line is represented as the x-axis and the vertical line is represented as the y-axis.

The four quadrants created via the division of the Cartesian plane corresponding to their respective values are:

Both the axes intersect at a point, which is known as the Origin (o). The location of any point on the Cartesian plane is indicated by a pair of values, called Coordinates. They are represented as x and y.

Once we identify the coordinates, then we can easily calculate the distance between the two points. Also, we can find the midpoint of that interval, which connects the points.

Representation of Equation of a Line on a Cartesian Plane

The equation of a line can be represented in several ways. 

We represent a general form of line as: 

Ax + By + C = 0.

Slope intercept Form 

To represent this way, let x & y be the coordinate of a point, through which a line passes. Now, m be the slope of a line, and c be the y-intercept. Then the equation of a line can be represented as :

y = mx + c

Slope of a Line

The general form of a line is Ax + By + C = 0. Now, the slope of this line can be found by converting this general form into the slope-intercept form.

Ax + By + C = 0 

⇒ By = − Ax – C

or,

y = -(Ax/B) – (C/B)

Here, on comparing the above-formed equation with the slope-intercept form, i.e., y = mx + c, we get,

m = -A/B

Hence, we can find the slope of a line directly from the general equation of a line.

Formulas and Theorems in Coordinate Geometry

Here are some essential formulas covered in class 10 maths Coordinate Geometry –

Distance Formula : 

Let the two points be A and B. These have coordinates (x1,y1) and (x2,y2), respectively.

The distance formula is:  

d = √((x2 – x1)2+(y2-y1)2)

Students can use this to find the distance between two points.

Mid-point Theorem

It is useful to find the midpoint of a line connecting two points.

Again assume two points A and B, having coordinates (x1,y1) and (x2,y2) respectively. Now, let M(x,y) be the midpoint lying on the line connecting points A and B. The pair of coordinates of the point M will be:

 M(x,y) = [(x1+x2)/2, (y1+y2)/2]

Section Formula

Students can use this to find a point that divides a line into an l:m ratio.

Let the line AB have coordinates (x1,y1) and (x2,y2). Now, assume D as a point that divides the line in the ratio l:m. Then the coordinates of the point P are given as-

If the ratio l:m is internal:

 [(lx2+mx1)/l+m, (ly2+my1)/l+m]

If the ratio l:m is external:

[(lx2-mx1)/l-m, (ly2-my1)/l-m]

Area of a Triangle in Cartesian Plane

In coordinate geometry, students can find the area of a triangle by this formula.

Let (x1,y1), (x2,y2), and (x3,y3) be the three vertices. Now, area of triangle =

 1/2 * [ x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2) ]

The three points are collinear if the area of a triangle with vertices are (x1,y1), (x2,y2), and (x3,y3) is zero.

Now, it is time for some examples. Consider a small exercise below.

1. Find the distance between the two points A(1, 2) and B(-2, 3).

Sol: Given two points, J (1, 2) and K (-2, 3).

Let J(1, 2) = (x1, y1) and K(-2, 3) = (x2, y2)

To calculate the distance between J and K using the distance formula,

D = √[(x2 – x1)2 + (y2 – y1)2]

JK = √[(-2 – 1)2 + (3 – 2)2]

JK = √(-3)2 + (1)2

JK = √9+1 = 3√1 unit.

2. Find the equation of the straight line passing through (2, 3) and is perpendicular to the line 3x + 2y + 4 = 0.

Sol: The given line is 3x + 2y + 4 = 0 

=> y = -3x / 2 – 2

Any line perpendicular to it will have slope = 2/3

So, equation of line through (2, 3) and slope 2 / 3 will be – 

(y – 3) = 2 / 3 (x – 2)

=> 3y – 9 = 2x – 4

=> 3y – 2x – 5 = 0

Hence, the equation is 3y – 2x – 5 = 0.

3. Find the area of the triangle formed by the vertices (5,6), (2,4), and (1,-3).

Sol: 1/2* [5(4 – (-3)) + 2(-3 – 6) + 1(6 – 4)]

       1/2* [5(7) +2(-9) + 1(2)]

       1/2* [35 – 18 + 2]

       1/2* [19] = 9.5

Hence, the answer is 9.5 square units.

4. At what point does the line 3x + y = -6 intercept the x-axis?

Sol: The line 3x + y = -6 will intercept x-axis at y = 0. 

Substituting this value of y in the above equation. 

3x + 0 = -6

x = -6/3 = -2

Hence, the line will intercept the x-axis at (-2,0).

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Conclusion

Coordinate Geometry is that branch of geometry where we determine the position of a point using coordinates. Coordinate Geometry can be a slightly confusing topic for some students because it involves formulas and algebraic calculations in the cartesian plane. However, we have explained the vital concepts of this topic here. Students can get full support, including study notes and doubt clearing sessions at our platform. We suggest that students should do regular practice applying these formulas to sums to gain a good command of the topic. Our notes can be of great help to recall the concepts whenever required. 

FAQs

1. What is the need for Coordinate geometry?

Coordinate geometry is needed to establish a connection between algebra and geometry by using graphs of lines and curves. It is the study of algebraic equations on graphs and is an important branch of mathematics. Typically, it helps locate points in a Cartesian plane. It is done by an ordered pair of numbers called Coordinates. With coordinates, we can locate any point on the plane. Notably, it has many uses in the fields of calculus, trigonometry, dimensional geometry, and more.

2. What are Quadrants, and how to locate them?

We can define Quadrants as the parts of a cartesian plane that are formed when the two axes intersect each other. When the axes intersect, we get four quadrants. To locate the four quadrants, you just have to move anti-clockwise. Quadrant I is the upper right region of the Cartesian plane, Quadrant II is the upper left region, Quadrant III is the lower-left region, and the fourth quadrant, Quadrant IV is the lower right region.

3. What is the distance formula? How is it used to calculate distance?

We know that when we join the given two points or coordinates, the distance between the two points is the length of the line segment. Such coordinates could lie on the x-axis or y-axis or both. We can calculate the distance using the formula derived from the Pythagoras theorem. The distance formula is √[(x2 – x1)2 + (y2 – y1)2]. Now, Assume (x1, y1) and (x2, y2) as the two points. Then, the distance between them will be = √[(x2 – x1)2 + (y2 – y1)2]. 

4. What is the significance of finding the coordinates?

If the students can determine the coordinates of a group of points, they can easily find the distance between these points; the equation, find the midpoint, and slope of the line segment; find whether the given lines are perpendicular or parallel; determine the area and perimeter of the polygon which formed by the points on the plane; find the equations of ellipses, curves, and circles; and transform the shape by moving, reflecting, and rotating it.

5. Are coordinates applicable in real life?

Yes, coordinates have real-life applications, such as the latitude and longitude lines on maps of the Earth are an example of spherical coordinates. The r-coordinate is located at the radius of the Earth, and the two-dimensional latitude and longitude planes are used to determine the location of different places on the surface of the Earth. Also, Air traffic is managed and controlled using coordinate geometry. The navigation system uses coordinates to ascertain the current location of the aircraft.

6. Is Coordinate Geometry an important topic for the competitive exams?

Coordinate geometry is a crucial topic for the national-level JEE Advanced 2022 Exam and JEE Main 2022 Exam. It is one of the undemanding and scoring topics of the JEE 2022. It has a considerable weightage of around 17-20% in the Mathematics section. It offers great scope for candidates to obtain a high rank in JEE Main 2022. The sub-topics in Coordinate Geometry are – Circle, Ellipse, Parabola, Hyperbola, and Straight Lines. The entire Coordinate geometry is important, but most questions are from straight lines and circles.