Class 11 Mathematics Notes Chapter 10 (Straight lines) – Mathematics Book

Alright class, let's get started with Chapter 10: Straight Lines from your NCERT Class 11 Mathematics textbook. This is a very important chapter, forming the bedrock of coordinate geometry, and concepts learned here are frequently tested in various government examinations. Pay close attention to the formulas and their applications.

Chapter 10: Straight Lines - Detailed Notes for Government Exam Preparation

1. Introduction & Basic Concepts

• Coordinate Geometry: Locating points in a plane using an ordered pair of numbers (x, y) with reference to two perpendicular lines (x-axis and y-axis).
• Distance Formula: The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by:
  PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Section Formula: Coordinates of a point R dividing the line segment joining P(x₁, y₁) and Q(x₂, y₂)
  • Internally in the ratio m:n are ( (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n) )
  • Externally in the ratio m:n are ( (mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n) )
  • Mid-point: ( (x₁ + x₂)/2, (y₁ + y₂)/2 ) (Ratio 1:1)
• Area of a Triangle: Area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:
  Area = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
  • Collinearity: Three points are collinear if the area of the triangle formed by them is zero.

2. Slope (Gradient) of a Line

• Definition: If θ is the angle of inclination of a non-vertical line (angle made with the positive x-axis, measured counterclockwise, 0° ≤ θ < 180°), then its slope 'm' is given by m = tan θ.
• Slope of a line passing through two points: If a line passes through P(x₁, y₁) and Q(x₂, y₂), its slope is m = (y₂ - y₁)/(x₂ - x₁) (where x₁ ≠ x₂).
• Special Cases:
  • Slope of the x-axis (or any horizontal line) is 0 (since θ = 0°).
  • Slope of the y-axis (or any vertical line) is undefined (since θ = 90°, tan 90° is undefined).
• Condition for Parallel Lines: Two non-vertical lines L₁ and L₂ are parallel if and only if their slopes are equal (m₁ = m₂).
• Condition for Perpendicular Lines: Two non-vertical lines L₁ and L₂ are perpendicular if and only if the product of their slopes is -1 (m₁ * m₂ = -1). Equivalently, m₂ = -1/m₁.
• Angle between Two Lines: The acute angle (say φ) between two non-vertical, non-perpendicular lines with slopes m₁ and m₂ is given by:
  tan φ = |(m₂ - m₁)/(1 + m₁m₂)|, provided 1 + m₁m₂ ≠ 0.
  • If 1 + m₁m₂ = 0, the lines are perpendicular.

3. Various Forms of the Equation of a Line

• Horizontal Line: Equation is y = k (where k is the distance from the x-axis).
• Vertical Line: Equation is x = k (where k is the distance from the y-axis).
• (a) Point-Slope Form: Equation of a line passing through a point (x₁, y₁) with slope 'm' is:
  y - y₁ = m(x - x₁)
• (b) Two-Point Form: Equation of a line passing through two points (x₁, y₁) and (x₂, y₂) is:
  (y - y₁) = [(y₂ - y₁)/(x₂ - x₁)] * (x - x₁)
• (c) Slope-Intercept Form: Equation of a line with slope 'm' and y-intercept 'c' is:
  y = mx + c
• (d) Intercept Form: Equation of a line making intercepts 'a' on the x-axis and 'b' on the y-axis is:
  x/a + y/b = 1 (where a ≠ 0, b ≠ 0)
• (e) Normal (or Perpendicular) Form: Equation of a line where 'p' is the length of the perpendicular from the origin to the line, and 'ω' (omega) is the angle this perpendicular makes with the positive x-axis is:
  x cos ω + y sin ω = p (where p > 0, 0° ≤ ω < 360°)

4. General Equation of a Line

• Any equation of the form Ax + By + C = 0, where A and B are not simultaneously zero, represents a straight line.
• Slope: If B ≠ 0, slope m = -A/B.
• Y-intercept: If B ≠ 0, y-intercept c = -C/B.
• X-intercept: If A ≠ 0, x-intercept a = -C/A.
• Conversion to Slope-Intercept Form: y = (-A/B)x + (-C/B) (assuming B ≠ 0)
• Conversion to Intercept Form: x/(-C/A) + y/(-C/B) = 1 (assuming A, B, C ≠ 0)
• Conversion to Normal Form:
  1. Move C to the RHS: Ax + By = -C.
  2. Make RHS positive: If -C is negative, multiply the whole equation by -1. Let the equation be A'x + B'y = C' (where C' is positive).
  3. Divide by √(A'² + B'²):
     (A'/√(A'²+B'²))x + (B'/√(A'²+B'²))y = C'/√(A'²+B'²)
  4. This is x cos ω + y sin ω = p, where p = C'/√(A'²+B'²), cos ω = A'/√(A'²+B'²), sin ω = B'/√(A'²+B'²). The quadrant of ω is determined by the signs of cos ω and sin ω.

5. Distance Formulas Involving Lines

• Distance of a Point from a Line: The perpendicular distance 'd' of a point P(x₁, y₁) from the line Ax + By + C = 0 is given by:
  d = |Ax₁ + By₁ + C| / √(A² + B²)
• Distance Between Two Parallel Lines: The distance 'd' between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is given by:
  d = |C₁ - C₂| / √(A² + B²)
  • Important: Ensure the coefficients of x and y are identical before applying this formula.

6. Family of Lines (Brief Overview)

• The equation of the family of lines passing through the intersection of two lines L₁ ≡ A₁x + B₁y + C₁ = 0 and L₂ ≡ A₂x + B₂y + C₂ = 0 is given by:
  L₁ + λL₂ = 0 => (A₁x + B₁y + C₁) + λ(A₂x + B₂y + C₂) = 0, where λ is a parameter.
  A specific line from this family can be found if one more condition (like passing through a given point, or having a specific slope) is provided, which helps determine λ.

7. Shifting of Origin

• If the origin (0, 0) is shifted to a new point (h, k) without rotating the axes, and the coordinates of a point P are (x, y) with respect to the old axes and (X, Y) with respect to the new axes, then the relationship is:
  x = X + h
y = Y + k
  Or equivalently:
  X = x - h
Y = y - k
• The equation of a curve changes when the origin is shifted.

Key Takeaways for Exams:

• Memorize all forms of the equation of a line and know when to use which form based on the given information.
• Be comfortable with slope calculations and the conditions for parallel and perpendicular lines.
• Master the distance formulas (point from line, between parallel lines).
• Practice converting the general equation into other forms, especially the normal form.
• Understand how to find the angle between two lines.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed. Try solving them:

1. The slope of the line passing through the points (2, 7) and (4, 9) is:
   a) 1
   b) 2
   c) -1
   d) 1/2

2. The equation of a line passing through the point (1, 2) with a slope of 3 is:
   a) y = 3x - 1
   b) y = 3x + 1
   c) y = 2x + 1
   d) y = x + 3

3. The equation of the line perpendicular to the line 2x + 3y + 5 = 0 and passing through (1, 1) is:
   a) 3x - 2y - 1 = 0
   b) 3x - 2y + 1 = 0
   c) 2x + 3y - 5 = 0
   d) 2x - 3y + 1 = 0

4. The intercepts made by the line 4x - 5y = 20 on the coordinate axes are:
   a) a = 5, b = 4
   b) a = 4, b = -5
   c) a = 5, b = -4
   d) a = -5, b = 4

5. The acute angle between the lines y = √3 x + 1 and y = (1/√3) x - 2 is:
   a) 30°
   b) 45°
   c) 60°
   d) 90°

6. The perpendicular distance of the point (3, -5) from the line 3x - 4y - 26 = 0 is:
   a) 3/5
   b) 7/5
   c) 0
   d) 1

7. The value of 'k' for which the lines 5x + 7y = 3 and 15x + ky = 9 are parallel is:
   a) 15
   b) 21
   c) 7
   d) 5

8. The normal form of the equation √3 x + y - 8 = 0 is:
   a) x cos 60° + y sin 60° = 4
   b) x cos 30° + y sin 30° = 8
   c) x cos 30° + y sin 30° = 4
   d) x cos 60° + y sin 60° = 8

9. The distance between the parallel lines 3x - 4y + 7 = 0 and 3x - 4y + 5 = 0 is:
   a) 2
   b) 2/5
   c) 12/5
   d) 1

10. The slope and y-intercept of the line 2x - 3y + 6 = 0 are:
    a) m = 2/3, c = 2
    b) m = -2/3, c = 2
    c) m = 2/3, c = -2
    d) m = 3/2, c = 3

Answers to MCQs:

1. a) 1 ( (9-7)/(4-2) = 2/2 = 1 )
2. a) y - 2 = 3(x - 1) => y - 2 = 3x - 3 => y = 3x - 1
3. a) Slope of given line = -2/3. Slope of perpendicular line = 3/2. Equation: y - 1 = (3/2)(x - 1) => 2y - 2 = 3x - 3 => 3x - 2y - 1 = 0
4. c) Divide by 20: (4x/20) - (5y/20) = 1 => x/5 + y/(-4) = 1. So a = 5, b = -4.
5. a) m₁ = √3, m₂ = 1/√3. tan φ = |(1/√3 - √3) / (1 + √3 * 1/√3)| = |((1-3)/√3) / (1+1)| = |(-2/√3) / 2| = |-1/√3| = 1/√3. So φ = 30°.
6. a) d = |3(3) - 4(-5) - 26| / √(3² + (-4)²) = |9 + 20 - 26| / √(9 + 16) = |3| / √25 = 3/5.
7. b) For parallel lines, A₁/A₂ = B₁/B₂. So, 5/15 = 7/k => 1/3 = 7/k => k = 21.
8. c) √3 x + y = 8. Divide by √( (√3)² + 1² ) = √(3+1) = √4 = 2. (√3/2)x + (1/2)y = 4. cos 30° = √3/2, sin 30° = 1/2. So, x cos 30° + y sin 30° = 4.
9. b) d = |7 - 5| / √(3² + (-4)²) = |2| / √25 = 2/5.
10. a) 3y = 2x + 6 => y = (2/3)x + 2. So m = 2/3, c = 2.

Remember to thoroughly revise these concepts and practice a wide variety of problems. Understanding the derivation and application of each formula is key to success. Good luck with your preparation!