Class 11 Mathematics Notes Chapter 10 (Chapter 10) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 10 from your NCERT Mathematics Exemplar book, which deals with Straight Lines. This is a foundational chapter in coordinate geometry and crucial for many competitive government exams. Pay close attention to the concepts and formulas.

Chapter 10: Straight Lines - Detailed Notes for Exam Preparation

1. Introduction & Basic Concepts Recap

• Coordinate Geometry: Locating points in a plane using ordered pairs (x, y).
• 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:
  • Internal Division: Coordinates of a point R dividing the line segment joining P(x₁, y₁) and Q(x₂, y₂) internally in the ratio m:n are:
    R = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)]
  • External Division: Coordinates of a point R dividing the line segment joining P(x₁, y₁) and Q(x₂, y₂) externally in the ratio m:n are:
    R = [(mx₂ - nx₁)/(m - n), (my₂ - ny₁)/(m - n)]
  • Mid-point Formula: (Special case of internal division where m:n = 1:1)
    Mid-point = [(x₁ + x₂)/2, (y₁ + y₂)/2]
• 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)|
  • Condition for Collinearity: Three points are collinear if the area of the triangle formed by them is zero.

2. Slope (Gradient) of a Line

• Definition: The slope 'm' of a non-vertical line is the tangent of the angle θ (inclination) it makes with the positive direction of the x-axis.
  m = tan θ (where 0° ≤ θ < 180°, θ ≠ 90°)
• Slope Formula (using two points): If a line passes through P(x₁, y₁) and Q(x₂, y₂), its slope is:
  m = (y₂ - y₁)/(x₂ - x₁) , provided x₁ ≠ x₂
• Nature of Slope:
  • Positive slope: Line rises from left to right (acute angle θ).
  • Negative slope: Line falls from left to right (obtuse angle θ).
  • Zero slope (m=0): Line is horizontal (parallel to x-axis, θ = 0°).
  • Undefined slope: Line is vertical (parallel to y-axis, θ = 90°).
• Parallel Lines: Two non-vertical lines are parallel if and only if their slopes are equal (m₁ = m₂).
• Perpendicular Lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (m₁ * m₂ = -1).
  • Note: A horizontal line (m=0) is perpendicular to a vertical line (undefined slope).

3. Angle Between Two Lines

• If θ is the acute angle between two lines with slopes m₁ and m₂, then:
  tan θ = |(m₂ - m₁)/(1 + m₁m₂)| , provided 1 + m₁m₂ ≠ 0.
• If 1 + m₁m₂ = 0, the lines are perpendicular (θ = 90°).
• If m₁ = m₂, the lines are parallel (θ = 0°).

4. Various Forms of the Equation of a Straight Line

Understanding these forms and when to use them is key:

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

5. General Equation of a Line

• Any equation of the form Ax + By + C = 0, where A and B are not both zero, represents a straight line.
• Converting General Form:
  • To Slope-Intercept Form: y = (-A/B)x + (-C/B)
    Slope (m) = -A/B
    Y-intercept (c) = -C/B (provided B ≠ 0)
  • To Intercept Form: x/(-C/A) + y/(-C/B) = 1
    X-intercept (a) = -C/A
    Y-intercept (b) = -C/B (provided A, B, C ≠ 0)
  • To Normal Form: Divide Ax + By + C = 0 by ±√(A² + B²) (choose sign such that the constant term becomes negative, i.e., -C/±√(A² + B²) = p > 0).
    (A/√(A² + B²))x + (B/√(A² + B²))y = -C/√(A² + B²)
    Here, cos ω = A/√(A² + B²), sin ω = B/√(A² + B²), p = |-C|/√(A² + B²) (adjust signs based on quadrant of ω).

6. 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:
  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:
  d = |C₁ - C₂| / √(A² + B²)
  • Important: Ensure the coefficients of x and y are identical before applying this formula.

7. Family of Lines

• The equation of any line passing through the point of intersection of two lines L₁: A₁x + B₁y + C₁ = 0 and L₂: A₂x + B₂y + C₂ = 0 is given by:
  L₁ + kL₂ = 0 => (A₁x + B₁y + C₁) + k(A₂x + B₂y + C₂) = 0
  where k is an arbitrary constant (parameter). The specific value of k is found using an additional condition (e.g., the line passes through another given point, or has a specific slope).

Exam Tips:

• Memorize all the formulas accurately.
• Understand the geometric interpretation behind each formula and form of the equation.
• Practice converting between different forms of the line equation quickly.
• Be careful with signs, especially in the Normal form and distance formulas.
• Remember the conditions for parallel (m₁ = m₂) and perpendicular lines (m₁m₂ = -1).
• For finding the intersection point of two lines, solve their equations simultaneously.
• The concept of locus often involves finding the equation of a line satisfying certain geometric conditions.

Multiple Choice Questions (MCQs)

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

1. The slope of the line passing through the points (2, 7) and (-1, 4) is:
   (a) 1
   (b) -1
   (c) 3
   (d) 1/3

2. The equation of a line parallel to the y-axis and passing through the point (-3, 5) is:
   (a) y = 5
   (b) x = -3
   (c) y = -3
   (d) x = 5

3. The equation of the line with slope 2 and y-intercept -3 is:
   (a) y = -3x + 2
   (b) y = 2x - 3
   (c) y = 2x + 3
   (d) x = 2y - 3

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

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

6. The x-intercept and y-intercept of the line 2x - 5y + 10 = 0 are respectively:
   (a) 5, 2
   (b) -5, 2
   (c) 5, -2
   (d) -5, -2

7. If the lines 3x + ky - 8 = 0 and 2x + y + 5 = 0 are perpendicular, the value of k is:
   (a) 6
   (b) -6
   (c) 3/2
   (d) -2/3

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

9. The points (1, 1), (-2, 7), and (3, -3) are:
   (a) Vertices of an equilateral triangle
   (b) Vertices of an isosceles triangle
   (c) Collinear
   (d) Vertices of a right-angled triangle

10. The normal form of the equation √3x + 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

Answers to MCQs:

1. (a) 1 [m = (4-7)/(-1-2) = -3/-3 = 1]
2. (b) x = -3 [Line parallel to y-axis has equation x = constant]
3. (b) y = 2x - 3 [Using y = mx + c]
4. (a) 30° [m₁ = √3, m₂ = 1/√3. tan θ = |(√3 - 1/√3)/(1 + √3 * 1/√3)| = |(2/√3)/2| = 1/√3. So θ = 30°]
5. (a) 3/5 [d = |3(3) - 4(-5) - 26| / √(3² + (-4)²) = |9 + 20 - 26| / √25 = |3|/5 = 3/5]
6. (b) -5, 2 [Put y=0 => 2x = -10 => x = -5. Put x=0 => -5y = -10 => y = 2]
7. (b) -6 [Slope 1 = -3/k. Slope 2 = -2/1 = -2. For perpendicular, m₁m₂ = -1 => (-3/k)(-2) = -1 => 6/k = -1 => k = -6]
8. (a) x - y + 1 = 0 [Slope of given line = -1. Slope of required line = 1. Eq: y - 2 = 1(x - 1) => y - 2 = x - 1 => x - y + 1 = 0]
9. (c) Collinear [Slope AB = (7-1)/(-2-1) = 6/-3 = -2. Slope BC = (-3-7)/(3-(-2)) = -10/5 = -2. Slopes are equal, points are collinear]
10. (c) x cos 30° + y sin 30° = 4 [Divide by √(√3² + 1²) = √4 = 2. (√3/2)x + (1/2)y = 4. Here cos ω = √3/2, sin ω = 1/2 => ω = 30°. p = 4]

Study these notes thoroughly and practice problems from the Exemplar book. Understanding these concepts well will significantly help in your exams. Good luck!