Most Important Questions Class 10 Maths Coordinate Geometry with Solutions

Coordinate geometry is one of the most fundamental and essential chapters in Class 10 mathematics. With its diverse applications and the scope for solving a variety of problems, it has become an important topic for students aiming to score high in their exams. At Educart, we’ll explore everything you need to know about learning coordinate geometry in Important Questions of Coordinate Geometry Class 10, extra practice techniques, and strategies with the help of available resources like solution PDFs.

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Chapter 7 Coordinate Geometry: Important Questions

1. △ABC is a triangle such that AB:BC = 1:2. Point A lies on the y-axis and the coordinates of B and C are known.

Which of the following formula can DEFINITELY be used to find the coordinates of A?

i) Section formula

ii) Distance formula

a. only i)

b. only ii)

c. both i) and ii)

d. neither i) or ii)

Soln.

(c) both i) and ii)

Explanation: 

Section Formula:
The section formula can be used to find the coordinates of a point that divides a line segment internally in a given ratio.

• Here, the ratio AB:BC=1:2 can help us determine A's position relative to B and C, especially if we treat A, B, and C as collinear points.
• Given that A lies on the y-axis, we can use the section formula to confirm its y-coordinate while fixing the x-coordinate as 0.

Thus, the section formula can be used to find A's coordinates.

Distance Formula:

The distance formula can be used to calculate the distance between two points.

• Using the known coordinates of B and C, as well as the condition AB:BC=1:2, we can compute the distances AB and BC.
• By ensuring the point A lies on the y-axis, we can solve for the y-coordinate of A using the distance condition.

Thus, the distance formula can also be used to find A's coordinates.

2. Preeti and Arun are both driving to their respective offices from the same home. Preeti drives towards the east at an average speed of 30 km per hour for 12 minutes and then towards the south at an average speed of 60 km per hour for 3 minutes. Arun drives towards the west at an average speed of 30 km per hour for 4 minutes and then towards the north at an average speed of 45 km per hour for 4 minutes.

What is the straight-line distance between Preeti's office and Arun's office? Show your steps and represent the given scenario on the coordinate plane.

Soln.  

Preeti's final position

Distance traveled east for 12 minutes: = Speed × Time 

= 30×1/5

=6 km.

(6,0)

Distance traveled  south for 3 minutes: = Speed × Time 

= 60×1/20

=3 km.

(3,0)

Since she is driving south, this means she moves 3 km along the negative y-axis. So, her final position is: (6,−3)

Arun's final position

Distance traveled west for 4 minutes: = Speed × Time 

= 30×1/15

=2 km.

(-2,0)

Distance traveled north for 4 minutes: = Speed × Time 

= 45×1/20

=2 km.

(2,0)

Since he is driving north, this means he moves 3 km along the positive y-axis. So, his final position is: (−2,3)

d= √(x₂​−x₁​)²+(y_2​−y₁​)²

d= √(-2​-6)​)²+(3- (-3))²

d= √(-8​)²+(6)²

d= √64+36

d= √100

d= 10km

The straight-line distance between Preeti's office and Arun's office is 10 km.

3. A circle with centre O(2,-5) has a chord with end-points A(1, 2) and B. M(5, -2) is the point where the perpendicular to the chord from the centre touches AB.

Find the coordinates of point B. Show your steps with valid reasons.

Soln.  

The center of the circle is O(2,−5)

A(1,2) and B(x,y) are the endpoints of the chord ABABAB.

M(5,−2) is the midpoint of AB where the perpendicular from the center O touches the chord.

M= x1​+x2 / 2, y1​+y2 / 2

(5,-2)= 1+x / 2, 2+y / 2

1+x / 2 = 5

x=9

2+y / 2 = -2

y=−6

B(9,−6)

4. Raaji and Gagan are finding a treasure that is exactly on the straight line joining them. Raaji's location is at (-6, -5) and Gagan's location is at (10, 11). The distance from the treasure to Raaji's location is three times that of the distance to Gagan's location.

Find the coordinates of the location of the treasure. Show your steps.

Soln.  

RT=3⋅TG.

T(x,y)= ((mx₂​+nx₁)​/m+n, (my₂​+ny₁)m+n)

x= 3(10)+1(−6)​/3+1

x= 30−6/4

x= 24/4

x= 6

y= 3(11)+1(−5)​/3+1

y= 33−5/4

y= 28/4

y= 7

The coordinates of the treasure are:

T(6,7)

5. On a golf course, three holes A(-6, -1), B and C(9, -4) lie on a straight line in that order.

The distance between B and C is two times that between B and A.Rahul strikes the ball, which is at point P(2, 3), such that it goes in the hole B.

i) Find the coordinates of hole B.

ii) Find the shortest distance covered by the ball.Show your steps.

Soln.  

i) Find the coordinates of hole B.

BC=2×BA

B divides AC in the ratio 1:2 (from A to C)

B(x,y)= ((mx₂​+nx₁)​/m+n, (my₂​+ny₁)m+n)

B(x,y)= (1(9)+2(−6)​/1+2, 1(−4)+2(−1)/1+2)

B(−1,−2)

ii) Find the shortest distance covered by the ball.

d=√(x₂​−x₁​)²+(y₂​−y₁​)²

d=√(-1​−2​)²+((-2)−3)²

d=√(-3​)²+(-5)²

d=√9+25

d=√34

6. The three vertices of a rhombus PQRS are P(2, -3), Q(6, 5) and R(-2, 1).

a) Find the coordinates of the point where both the diagonals PR and QS intersect.

b) Find the coordinates of the fourth vertex S.Show your steps and give valid reasons.

Soln.  

The vertices of the rhombus are given as:

• P(2,−3),
• Q(6,5)
• R(−2,1)

Key Properties of a Rhombus:

1. The diagonals of a rhombus bisect each other at right angles.
2. This means the point of intersection of the diagonals is the midpoint of both diagonals.

a) Find the coordinates of the point where both the diagonals PR and QS intersect. 

M= (x₁​+x₂​)/2 , (y₁​+y₂​)/2 , 

M= (2​+(-2)​)/2 , (-3+1)/2 , 

M= 0/2 , (-2)/2 , 

M= 0 , -1 

Thus, the diagonals intersect at the point (0,−1)

b) Find the coordinates of the fourth vertex S.

M(0,−1) is the midpoint,

Q(6,5) and S(x,y) are the endpoints.

(6+x)/2, (5+y)/2=(0,−1)

For x:

(6+x)/2 = 0

6+x=0

x=−6

For y

(5+y)/2= -1

5+y=−2

y=−7

Thus, the coordinates of S are (−6,−7)

7. The line x + 2y =2 forms a triangle OPQ, with the coordinate axes.

(i) What are the coordinates of points P and Q?

(ii) What is the area of the triangle formed? Show your steps.

Soln.  

(i) What are the coordinates of points P and Q?

The given line is: 

x+2y=2

For the x-axis 

Substitute y=0 into x+2y=2

x+2(0)=2⟹x=2

So, Q has coordinates (2,0)

For the y-axis 

Substitute x=0 nto x+2y=2

0+2y=2⟹y=1

So, P has coordinates (0,1)

Thus, the coordinates of P and Q are:

P(0,1)and Q(2,0)

(ii) What is the area of the triangle formed? Show your steps.

O(0,0)

P(0,1),

Q(2,0)

area of a △ = ½ ∣x_1​(y₂​−y₃​)+x₂​(y₃​−y₁​)+x₃​(y₁​−y₂​)∣

area of a △ = ½ ∣0_​(1-0​)+0​(​0−0​)+02(0​−1​)∣

area of a △ = ½ ∣0_​+0​+2​(​−1​)∣

area of a △ = ½ ∣0_​+0​-2​∣

area of a △ = ½ ∣​-2​∣

area of a △ = ½ 2​

area of a △ = 1

8. A(5, 1), B(1, 4) and C(8, 5) are the coordinates of the vertices of a triangle.

Which of the following types of triangle will △ABC be?

a. Equilateral triangle

b. Isosceles right-angled triangle

c. Scalene right-angled triangle

d. Isosceles acute-angled triangle

Soln.  (b) Isosceles right-angled triangle

Explanation:

AB=√(x_2​−x₁​)²+(y₂​−y₁​)²

AB= √(1_​−5​)²+(4-1​)²

AB= √(4)²+(3)²

AB= √16+9

AB= √25

AB= 5

BC =√(x_2​−x₁​)²+(y₂​−y₁​)²

BC = √(8_​−1​)²+(5-4​)²

BC = √(7)²+(1)²

BC = √49+1

BC = √50

BC = 5√2

CA =√(x_2​−x₁​)²+(y₂​−y₁​)²

CA = √(8_​−5)²+(5-1)²

CA = √(3)²+(4)²

CA = √9+16

CA = √25

CA = 5

Two sides are equal: AB=CA=5

This makes the triangle isosceles.

Checking for a right angle:

BC²=AB²+CA²

(5√2​)²=5²+5²

(50​)=25 + 25

(50​)=50

The condition holds true, so the triangle is right-angled.

The triangle △ABC is an isosceles right-angled triangle.

9. The radius of a circle with centre at the origin is ½ units.

Find all the points on the circle which are of the form (-y, y ). Show your steps.

Soln.  

The equation of a circle with center at the origin (0,0) and radius r is:

x²+y²=r²

x²+y²=(½)²

x²+y²=(1/4)

x=−y

y=y

(−y)²+y²=¼

(y)²+y²=¼

2y²=¼

y²=¼ / 2

y²= 8

y= ± √2/4

(−y,y)= (−√2/4, √2/4)

(−y,y)= -(−(√2/4), -√2/4)

(−y,y)= (√2/4, -√2/4)

points on the circle of radius 1/2 that are of the form (−y,y) are:(−√2/4, √2/4) or (√2/4, -√2/4)

Introduction to Coordinate Geometry in Class 10

Coordinate geometry, often referred to as analytic geometry, is the study of geometry using a coordinate system. In Class 10, this topic deals majorly with the Cartesian plane, where students learn how to solve problems related to points, lines, distances, and areas using coordinates (x, y) on a two-dimensional plane.

The core concepts include:

• The distance formula
• The section formula
• Finding the coordinates for the midpoint
• Understanding the slope of a line
• Calculate the area of a triangle using its vertices on a coordinate plane.

As exams draw near, the important questions from Coordinate Geometry class 10 play a significant role in your preparation. We carefully curate these questions to cover the wide variety of concepts and problem-solving approaches you might encounter in your exams. Practicing these can not only build confidence but also provide insights into the kinds of questions that frequently appear in board exams.

Why are Important Questions in Coordinate Geometry an Essential Resource?

Class 10 is an important and tough year for students. The marks obtained in board exams are a reflection of your hard work and consistency. One of the best ways to ensure success is by practicing important questions in Coordinate Geometry Class 10.  But what makes these questions so important?

• The class 10 coordinate geometry questions are designed to cover the entire syllabus compactly. They target important areas where students typically face challenges, ensuring that you're focusing on the concepts that matter most.
• Important questions frequently mimic the kinds of problems you'll encounter in board exams. By practicing these questions, you familiarize yourself with the exam format, making it easier to recognize patterns and solve problems quickly during the actual exam.
• Practicing class 10 coordinate geometry with extra questions helps ensure you don't miss out on any topics.  These questions typically touch on every major concept, from simple distance and section formulas to more complex problems involving triangles and slopes.
• As you solve more of these Coordinate Geometry Class 10 Important Questions, your confidence grows. You'll find that your problem-solving speed increases and your understanding of the concepts deepens. This self-assurance will be your biggest help during exams.

• Class 10 exams aren’t just about memorizing formulas; they require application. We structure extra questions to test your problem-solving skills. The more you practice, the better equipped you'll be to handle any question.

How to Use Class 10 Important Questions with Solutions for Maximum Benefits

Effectively utilizing coordinate geometry in class 10 important questions is a crucial aspect of exam preparation. This detailed guide will help you maximize these resources:

• Use the extra questions and solutions to identify where you're struggling. If you find that you're consistently making errors in a particular type of question, dedicate additional time to that topic. You can revise the theory, practice more questions, and then review the solution PDFs to make sure you're on the right track.
• Once you have enough practice, it's a good idea to solve the important class 10 coordinate geometry questions under timed conditions. Set a timer for the same amount of time as an exam and solve the questions without solutions or notes. This will help you build exam discipline and improve time management.
• Regularly review the solutions to the questions you’ve solved. Sometimes, we think we’ve mastered a concept, only to realize later that we’ve forgotten the details. Every week, dedicate a day to review the solutions from the Coordinate Geometry Class 10 Extra Questions with Solutions PDF that you have practiced. This way, the concepts stay fresh in your mind.
• Coordinate geometry is a visual subject. Make use of graphs and diagrams while solving questions. Draw the coordinate plane, plot the points, and see the relationships between the geometric shapes and the coordinate system. This not only enhances understanding but also helps in retaining information longer.

How to Use Class 10 Important Questions Effectively?

To truly excel in the coordinate geometry chapter of Class 10, here are some unique tips:

• It’s tempting to simply memorize the formulas for distance, section, and area. However, understanding the derivation of these formulas can significantly improve your ability to apply them correctly in unfamiliar situations.
• When practicing, use graph paper. Plotting points and lines on graph paper makes the geometric relationships clearer and helps understand the nuances of questions better.
• Many questions in exams are word problems. Ensure you understand how to translate these word problems into coordinate geometry equations or diagrams.
• Take periodic self-tests using extra questions to check your progress. This will help you stay focused on your preparation and determine the additional work required in specific areas.
• Refer to past year's papers and sample papers to understand the latest trends in the types of questions asked. Occasionally, similar patterns or repetitions of questions occur.

Coordinate geometry class: 10 important questions play a major role in improving your understanding of this chapter and preparing you for the board exams. By practicing a diverse set of class 10 coordinate geometry extra questions, using available solutions PDFs, and following the tips, you’ll be well on your way to learning coordinate geometry and scoring high in your exams.

To top the exam, the key is consistent practice, understanding core concepts, and using resources like solution PDFs wisely. Make a study plan, stick to it, and don’t forget to review your mistakes.