🔹 1. Introduction

A straight line is one of the simplest geometrical figures in coordinate geometry. It represents a linear relationship between two variables.

👉 General form:

• A straight line equation connects x and y in first degree.

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🔹 2. Basic Concepts

✔️ Coordinate System

• Point in plane → (x, y)
• x-axis → horizontal
• y-axis → vertical

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✔️ Distance Formula

Distance between two points:

$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$d=(x2​−x1​)2+(y2​−y1​)2​-10-8-6-4-2246810-10-5510(6.0, 6.0)(-6.0, -6.0)d = 16.97

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✔️ Section Formula

Point dividing line segment in ratio m:n:

Internal Division:

$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$(m+nmx2​+nx1​​,m+nmy2​+ny1​​)

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✔️ Midpoint Formula

$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$(2×1​+x2​​,2y1​+y2​​)-10-8-6-4-2246810-10-5510A(-7.0, -3.0)B(5.0, 7.0)M = (-1.0, 2.0)

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🔹 3. Slope of a Line

✔️ Definition

Slope (m) measures steepness of a line.

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✔️ Formula

$m = \frac{y_2 – y_1}{x_2 – x_1}$m=x2​−x1​y2​−y1​​-10-8-6-4-2246810-10-5510-8.00, -8.008.00, 8.00m = 1.00

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✔️ Special Cases

• m > 0 → increasing line
• m < 0 → decreasing line
• m = 0 → horizontal line
• Undefined → vertical line

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🔹 4. Angle of Inclination

Angle (θ) between line and positive x-axis:

$m = \tan\theta$m=tanθ

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🔹 5. Equation of a Line

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✔️ 1. Slope-Intercept Form

$y = mx + c$y=mx+c

$m$m

$b$b-10-8-6-4-2246810-10-5510y-interceptx-intercept

• m = slope
• c = y-intercept

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✔️ 2. Point-Slope Form

$y – y_1 = m(x – x_1)$y−y1​=m(x−x1​)-10-8-6-4-2246810-10-5510-8.00, -8.008.00, 8.00m = 1.00

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✔️ 3. Two-Point Form

$y – y_1 = \frac{y_2 – y_1}{x_2 – x_1}(x – x_1)$y−y1​=x2​−x1​y2​−y1​​(x−x1​)-10-8-6-4-2246810-10-5510-8.00, -8.008.00, 8.00m = 1.00

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✔️ 4. Intercept Form

$\frac{x}{a} + \frac{y}{b} = 1$ax​+by​=1

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✔️ 5. General Form

$Ax + By + C = 0$Ax+By+C=0

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🔹 6. Conditions for Parallel and Perpendicular Lines

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✔️ Parallel Lines

Slopes are equal:

$m_1 = m_2$m1​=m2​

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✔️ Perpendicular Lines

Product of slopes:

$m_1 m_2 = -1$m1​m2​=−1

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🔹 7. Angle Between Two Lines

$\tan\theta = \left|\frac{m_1 – m_2}{1 + m_1 m_2}\right|$tanθ=​1+m1​m2​m1​−m2​​​

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🔹 8. Distance of a Point from a Line

Distance of point (x₁, y₁) from line Ax + By + C = 0:

$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$d=A2+B2​∣Ax1​+By1​+C∣​

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🔹 9. Family of Lines

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✔️ Through Intersection of Two Lines

$L_1 + \lambda L_2 = 0$L1​+λL2​=0

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✔️ Parallel Lines

$Ax + By + C = 0$Ax+By+C=0

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✔️ Perpendicular Lines

Slope = −1/m

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🔹 10. Special Cases

• x = constant → vertical line
• y = constant → horizontal line

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🔹 11. Important Results

✔️ Equation of x-axis → y = 0
✔️ Equation of y-axis → x = 0
✔️ Lines parallel to axes have simple forms

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🔹 12. Applications

✔️ Coordinate geometry problems
✔️ Physics (motion, vectors)
✔️ Engineering calculations
✔️ Graph interpretation

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🔹 13. JEE & CBSE Important Points

✔️ Master all forms of equation
✔️ Practice slope-based problems
✔️ Distance formula is very important
✔️ Angle between lines frequently asked
✔️ Family of lines important for JEE

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🔹 14. Common Mistakes

❌ Sign mistakes in slope
❌ Wrong substitution in formulas
❌ Confusing forms of equations
❌ Ignoring vertical line slope

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🔹 15. Practice Questions

1. Find slope of line joining two points
2. Find equation using point-slope form
3. Check if lines are perpendicular
4. Find distance of point from line
5. Find angle between two lines

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🔹 16. Quick Revision Sheet

• Slope = (y₂−y₁)/(x₂−x₁)
• y = mx + c
• Ax + By + C = 0
• Distance = |Ax₁ + By₁ + C| / √(A² + B²)
• Parallel → m₁ = m₂
• Perpendicular → m₁m₂ = −1

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🔹 17. Conclusion

Straight Lines is a fundamental chapter of coordinate geometry and forms the base for:

• Circles
• Conic Sections
• 3D Geometry

👉 Strong understanding ensures high scores in CBSE & JEE exams.