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I. Chapter Summary

Coordinate Geometry combines algebra and geometry to study the position of points, lines, and shapes on the Cartesian plane. This chapter introduces students to concepts such as distance between two points, section formula, and mid-point formula using coordinate geometry. These tools help in solving real-world geometric problems algebraically.

II. Key Concepts Covered

Concept	Explanation
Coordinate Plane	A two-dimensional plane formed by a horizontal (x-axis) and a vertical (y-axis) axis intersecting at the origin (0, 0).
Distance Formula
To find distance between two points $A(x_1, y_1) quad text{and} quad B(x_2, y_2)$
$ sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Section Formula
To find coordinates of point dividing line segment in ratio $m:n$
$left( frac{mx_2 + nx_1}{m+n}, frac{my_2 + ny_1}{m+n} right)$
Mid-Point Formula	Special case of section formula when $m=n$ :
$left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2} right)$
Collinearity of Points	If the sum of distances between two pairs equals the third, the points are collinear.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. What is the distance between the points (3, 4) and (0, 0)?
   a) 5 ✅ (PYQ 2020)
   b) 4
   c) 3
   d) 6
2. The coordinates of the mid-point of the line joining (2, 3) and (4, 7) are:
   a) (3, 5) ✅
   b) (2, 7)
   c) (6, 10)
   d) (1, 5)
3. If a point (x, y) divides the line joining (2, 4) and (6, 8) in the ratio 1:1, then (x, y) is:
   a) (3, 5)
   b) (4, 6) ✅
   c) (5, 7)
   d) (6, 8)
4. Which of the following is a correct use of distance formula?
   a) To find slope
   b) To find length between two points ✅
   c) To find angle between lines
   d) To find area of triangle

(B) Short Answer Questions (2/3 Marks)

1. Find the distance between the points A(1, 2) and B(4, 6).
2. Find the coordinates of the point that divides the line segment joining (2, -3) and (4, 1) in the ratio 3:1.
3. Find the mid-point of the line joining the points (-1, 2) and (3, 6).
4. Show that the points (1, 2), (3, 4), and (5, 6) are not collinear.

(C) Long Answer Questions (5 Marks)

1. Find the coordinates of the point which divides the line joining A(2, 3) and B(4, 5) in the ratio 2:3 internally. Also verify the distance between A and the point.
2. Using distance formula, prove that triangle ABC with vertices A(2, 3), B(6, 7), and C(2, 7) is a right-angled triangle.
3. Show that the triangle with vertices A(1, 1), B(4, 4), and C(1, 4) is an isosceles right triangle.
4. The line joining the points A(2, 3) and B(4, y) is of length 5 units. Find the value of y.

(D) HOTS (Higher Order Thinking Skills)

1. The mid-point of the line joining A(4, x) and B(10, 6) is (7, y). Find x and y.
2. If the point P(x, 2) is equidistant from A(2, -1) and B(4, 3), find the value of x.

IV. Key Formulas/Concepts

Formula	Use
Distance Formula	$d = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Section Formula	$left( frac{mx_2 + nx_1}{m+n}, frac{my_2 + ny_1}{m+n} right)$
Mid-Point Formula	$(x_1 + x_2, y_1 + y_2), quad left( frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2} right), quad (2x_1 + x_2, 2y_1 + y_2)$
Collinearity Condition	Points A, B, and C are collinear if $AB + BC = AC$

V. Deleted Portions (CBSE 2025–2026)

 No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)

Unit/Chapter	Estimated Marks	Type of Questions Typically Asked
Chapter 7: Coordinate Geometry	6 Marks	1 Long Answer, 1–2 Short Answers, 1 MCQ

VII. Previous Year Questions (PYQs)

Year	Marks	Question
2020	1	Find the distance between origin and point (3, 4)
2019	3	Mid-point and section formula-based word problem
2018	5	Use coordinate geometry to prove triangle is right-angled

VIII. Real-World Application Examples

Application	Explanation
GPS and Navigation	Uses coordinate geometry to calculate distances between locations.
Architecture & Design	Coordinates are used in blueprints and CAD tools.
Computer Graphics	Coordinate systems are the basis of game and animation design.
Surveying	Plotting land areas and planning using coordinate systems.

IX. Student Tips & Strategies for Success

Time Management

• Practice at least 2 coordinate geometry problems every day.
• Focus on understanding formulas rather than memorizing.

Exam Preparation

• Label diagrams correctly.
• Practice PYQs that involve proving triangle types using the distance formula.

Stress Management

• Use graph paper to visualize coordinates.
• Break down word problems into smaller steps.

X. Career Guidance & Exploration (Class-Specific)

For Classes 9–10

Streams & Careers

• Science: Engineering, Robotics, GIS, Astronomy
• Commerce: Data Analytics, Logistics
• Arts: Urban Planning, Architecture

Competitive Exams

• NTSE, CUET Quantitative, RMO, IMO

XI. Important Notes

• Always draw neat diagrams and mark coordinates clearly.
• Practice applying all three formulas: distance, section, and mid-point.
• Visit https://ncert.nic.in and https://cbseacademic.nic.in for updates.