Straight Lines — Previous Year Questions (Class 11 Mathematics)
Straight lines are fundamental in coordinate geometry. Master equations, slopes, and distances for problem solving.
TL;DR: Straight lines are fundamental in coordinate geometry. Master equations, slopes, and distances for problem solving.
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Straight lines are fundamental in coordinate geometry. Master equations, slopes, and distances for problem solving.
Straight Lines — Previous Year Questions with Solutions
Q (2023, 2 marks): Find the equation of the line passing through points (2, 3) and (4, 7).
Answer: Points: (x₁, y₁) = (2, 3), (x₂, y₂) = (4, 7)
Slope m = (y₂ - y₁)/(x₂ - x₁) = (7 - 3)/(4 - 2) = 4/2 = 2
Using point-slope form: y - y₁ = m(x - x₁)
y - 3 = 2(x - 2)
y - 3 = 2x - 4
y = 2x - 1
Or: 2x - y - 1 = 0
Final Answer: y = 2x - 1 or 2x - y - 1 = 0
Q (2022, 3 marks): Find the distance between the parallel lines 3x + 4y - 5 = 0 and 3x + 4y + 10 = 0.
Answer: For parallel lines ax + by + c₁ = 0 and ax + by + c₂ = 0:
Distance = |c₁ - c₂|/√(a² + b²)
Here: a = 3, b = 4, c₁ = -5, c₂ = 10
Distance = |-5 - 10|/√(3² + 4²) = |-15|/√(9 + 16) = 15/√25 = 15/5 = 3
Final Answer: Distance = 3 units
Q (2023, 3 marks): Find the angle between the lines 2x - y + 3 = 0 and x + 2y - 5 = 0.
Answer: Line 1: 2x - y + 3 = 0, slope m₁ = 2
Line 2: x + 2y - 5 = 0, or y = -x/2 + 5/2, slope m₂ = -1/2
Angle between lines: tan(θ) = |(m₁ - m₂)/(1 + m₁m₂)|
tan(θ) = |(2 - (-1/2))/(1 + 2(-1/2))|
tan(θ) = |(2 + 1/2)/(1 - 1)|
tan(θ) = |(5/2)/0| = undefined
When denominator is 0, the angle is 90°.
Verification: m₁ × m₂ = 2 × (-1/2) = -1, confirming perpendicular lines.
Final Answer: θ = 90° (lines are perpendicular)
Q (2021, 3 marks): Find the equation of the line perpendicular to 3x + 4y - 12 = 0 and passing through (1, 2).
Answer: Given line: 3x + 4y - 12 = 0
Slope of given line: m₁ = -3/4
Slope of perpendicular line: m₂ = -1/m₁ = -1/(-3/4) = 4/3
Equation of perpendicular line passing through (1, 2):
y - 2 = (4/3)(x - 1)
3(y - 2) = 4(x - 1)
3y - 6 = 4x - 4
4x - 3y + 2 = 0
Final Answer: 4x - 3y + 2 = 0
Q (2022, 2 marks): Find the distance of the point (3, 4) from the line 4x - 3y + 2 = 0.
Answer: Distance of point (x₀, y₀) from line ax + by + c = 0:
Distance = |ax₀ + by₀ + c|/√(a² + b²)
Point: (3, 4), Line: 4x - 3y + 2 = 0
Distance = |4(3) - 3(4) + 2|/√(4² + (-3)²)
Distance = |12 - 12 + 2|/√(16 + 9)
Distance = |2|/√25 = 2/5 = 0.4
Final Answer: Distance = 2/5 units or 0.4 units
Q (2023, 3 marks): Find the equation of the line making an angle of 45° with the positive x-axis and passing through (2, 3).
Answer: Angle with positive x-axis = 45°
Slope m = tan(45°) = 1
Point: (2, 3)
Using point-slope form:
y - 3 = 1(x - 2)
y - 3 = x - 2
y = x + 1
Or: x - y + 1 = 0
Final Answer: y = x + 1 or x - y + 1 = 0
Frequently Asked Questions
What are the different forms of equation of a line?
1. Slope-intercept form: y = mx + c. 2. Point-slope form: y - y₁ = m(x - x₁). 3. Standard form: ax + by + c = 0. 4. Intercept form: x/a + y/b = 1. 5. Parametric form: x = x₁ + lt, y = y₁ + mt. Choose based on given information.
How do you determine if two lines are parallel or perpendicular?
For lines with slopes m₁ and m₂: Parallel if m₁ = m₂ (same slope). Perpendicular if m₁ × m₂ = -1 (product is -1). For lines ax₁ + by₁ + c₁ = 0 and ax₂ + by₂ + c₂ = 0: Parallel if a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Perpendicular if a₁a₂ + b₁b₂ = 0.
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