Coordinate Geometry – Complete Notes for Competitive Exams

1. Introduction

Coordinate Geometry, also called Analytical Geometry, deals with geometric figures on the Cartesian plane using coordinates $(x, y)$(x,y).

It is a crucial topic in SSC, Banking, Railways, and other competitive exams.

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2. Basic Concepts

• Point: Represented as $P(x, y)$P(x,y)
• Distance between two points: $d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$d=(x2​−x1​)2+(y2​−y1​)2​
• Midpoint of two points: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$M=(2×1​+x2​​,2y1​+y2​​)
• Slope of a line: $m = \frac{y_2 – y_1}{x_2 – x_1}$m=x2​−x1​y2​−y1​​
• Equation of a line:
  • Slope-intercept form: $y = mx + c$y=mx+c
  • Point-slope form: $y – y_1 = m(x – x_1)$y−y1​=m(x−x1​)
• Distance from point to line: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$d=A2+B2​∣Ax1​+By1​+C∣​

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3. Equations of Standard Lines

1. Horizontal line: $y = k$y=k
2. Vertical line: $x = h$x=h
3. Line through two points: $y – y_1 = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1)$y−y1​=x2​−x1​y2​−y1​​(x−x1​)
4. Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ax​+by​=1

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4. Important Formulas

• Slope of line joining points: $m = \frac{y_2 – y_1}{x_2 – x_1}$m=x2​−x1​y2​−y1​​
• Slope of perpendicular lines: $m_1 \cdot m_2 = -1$m1​⋅m2​=−1
• Slope of parallel lines: $m_1 = m_2$m1​=m2​
• Area of triangle formed by points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$(x1​,y1​),(x2​,y2​),(x3​,y3​):

$$A = \frac{1}{2} |x_1(y_2 – y_3) + x_2(y_3 – y_1) + x_3(y_1 – y_2)|$$A=21​∣x1​(y2​−y3​)+x2​(y3​−y1​)+x3​(y1​−y2​)∣

• Collinearity of points: Area = 0

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5. Important Tips

• Always plot points for clarity
• Use distance formula for lengths
• Use slope formula to check parallel or perpendicular lines
• For triangles, check area formula or determinant method

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Top 25 Practice Questions – Coordinate Geometry

Points and Distance

Q1. Find distance between points (3, 4) and (7, 1)
Q2. Find midpoint of points (2, 3) and (6, 7)
Q3. Find slope of line joining points (1, 2) and (3, 6)
Q4. Check if points (1, 2), (3, 6), (5, 10) are collinear
Q5. Find distance of point (4, 3) from line y = 2x + 1

Lines and Slopes

Q6. Equation of line passing through (2, 3) with slope 4
Q7. Equation of line passing through points (1, 2) and (3, 6)
Q8. Find slope of line perpendicular to y = 3x + 4
Q9. Find slope of line parallel to y = -2x + 5
Q10. Equation of line passing through (0, 0) and perpendicular to y = x + 1

Triangles and Area

Q11. Area of triangle formed by points (0, 0), (4, 0), (0, 3)
Q12. Check if points (1, 1), (2, 3), (3, 5) form a triangle
Q13. Area of triangle with points (2, 3), (5, 7), (8, 1)
Q14. Find equation of median from vertex (2, 3) of triangle with other vertices (4, 7), (6, 1)
Q15. Find centroid of triangle with vertices (0, 0), (6, 0), (0, 8)

Advanced

Q16. Equation of line passing through intersection of lines y = 2x + 3 and y = -x + 5
Q17. Distance between points on x-axis (a, 0) and y-axis (0, b)
Q18. Find slope of line joining midpoint of (1, 2) & (3, 4) to midpoint of (5, 6) & (7, 8)
Q19. Find equation of line passing through centroid of triangle (0,0), (4,0), (0,3)
Q20. Find distance from point (2, 3) to line 3x – 4y + 5 = 0
Q21. Check if line joining (1, 1) & (4, 5) is perpendicular to line joining (2, 3) & (5, 1)
Q22. Equation of perpendicular bisector of line joining points (1, 2) and (5, 6)
Q23. Find coordinates of point dividing line joining (2, 3) & (6, 7) in ratio 2:1
Q24. Area of triangle formed by points (1, 1), (4, 5), (7, 2)
Q25. Find distance of point (3, 4) from y-axis

Answer

Answers – Coordinate Geometry

Q1. 5 units
Q2. (4, 5)
Q3. 2
Q4. Yes, collinear
Q5. 0.894 units
Q6. y – 3 = 4(x – 2) → y = 4x – 5
Q7. y – 2 = 2(x – 1) → y = 2x
Q8. -1/3
Q9. -2
Q10. y = -x
Q11. 6 units²
Q12. No, they are collinear
Q13. 12 units²
Q14. Equation of median: y – 3 = -0.5(x – 2)
Q15. (2, 8/3)
Q16. y = -1/3 x + 2 (example)
Q17. √(a² + b²)
Q18. 1
Q19. Line through centroid: y – 11/3 = ? (example)
Q20. 2.17 units
Q21. Yes, slopes multiply to -1
Q22. x + y – 6 = 0
Q23. (10/3, 5/3)
Q24. 10.5 units²
Q25. 3 units