Straight Lines and Coordinate Geometry Solved Examples (Class 11 Mathematics)
Straight lines numericals cover equations of lines, slopes, distances, and angle relationships. These problems develop analytical geometry skills essential
TL;DR: Straight lines numericals cover equations of lines, slopes, distances, and angle relationships. These problems develop analytical geometry skills esse…
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Straight lines numericals cover equations of lines, slopes, distances, and angle relationships. These problems develop analytical geometry skills essential
Straight Lines and Coordinate Geometry — Solved Numerical Examples (Step by Step)
Example 1: Find the slope of the line passing through points (2, 3) and (5, 9).
Solution: Points: (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 9). Slope m = (y₂ - y₁) / (x₂ - x₁) = (9 - 3) / (5 - 2) = 6 / 3 = 2.
Example 2: Find the equation of line with slope 3 and y-intercept 5.
Solution: Using y = mx + c where m = 3 and c = 5. Equation: y = 3x + 5 or 3x - y + 5 = 0.
Example 3: Find the distance between points (1, 2) and (4, 6).
Solution: Points: (x₁, y₁) = (1, 2) and (x₂, y₂) = (4, 6). Distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²) = sqrt((4-1)² + (6-2)²) = sqrt(9 + 16) = sqrt(25) = 5.
Example 4: Find the angle between lines with slopes 2 and -1/2.
Solution: Slopes m₁ = 2, m₂ = -1/2. Using tan(θ) = |(m₁ - m₂) / (1 + m₁m₂)| = |(2 - (-1/2)) / (1 + 2 × (-1/2))| = |(2.5) / 0| which is undefined, indicating perpendicular lines (90 degrees). Verify: m₁ × m₂ = 2 × (-1/2) = -1, confirming perpendicularity.
Example 5: Find the perpendicular distance from point (1, 2) to the line 3x + 4y - 6 = 0.
Solution: Line equation: 3x + 4y - 6 = 0. Point: (x₀, y₀) = (1, 2). Distance = |ax₀ + by₀ + c| / sqrt(a² + b²) = |3(1) + 4(2) - 6| / sqrt(9 + 16) = |3 + 8 - 6| / 5 = 5 / 5 = 1.
Example 6: Find the equation of line passing through (2, 3) with slope -2.
Solution: Point: (x₀, y₀) = (2, 3), Slope m = -2. Using point-slope form: y - y₀ = m(x - x₀). y - 3 = -2(x - 2). y - 3 = -2x + 4. y = -2x + 7 or 2x + y - 7 = 0.
Tips
- Slope m = (y₂ - y₁) / (x₂ - x₁).
- Perpendicular lines have slopes m₁ × m₂ = -1.
- Distance from point to line ax + by + c = 0 is |ax₀ + by₀ + c| / sqrt(a² + b²).
Frequently Asked Questions
What does slope represent?
Slope is the measure of steepness of a line. Positive slope means line goes up from left to right. Negative slope means line goes down. Slope = rise/run.
How can we determine if lines are parallel or perpendicular?
Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have product of slopes = -1 (m₁ × m₂ = -1).
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
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