Master the concept of slope of a line in CBSE Class 11 Applied Mathematics. Learn formulas, coordinate geometry applications, and solved examples.
The slope of a line, often denoted by m, represents the steepness and direction of a line in a Cartesian plane. It is defined as the tangent of the angle θ that the line makes with the positive direction of the x-axis. If a line passes through two distinct points (x₁, y₁) and (x₂, y₂), the slope is the ratio of the vertical change to the horizontal change.
Step 1: Identify points (2, 3) and (5, 9).
Step 2: Apply formula m = (9 - 3)/(5 - 2).
Step 3: Calculate m = 6/3.
Answer: m = 2
Two lines are parallel if their slopes are equal, meaning m₁ = m₂. Two lines are perpendicular if the product of their slopes is -1, provided neither line is vertical. This relationship is fundamental for determining the geometric properties of quadrilaterals and triangles.
Step 1: Line 1 has slope m₁ = 2. Line 2 has slope m₂ = -0.5.
Step 2: Multiply m₁ · m₂ = 2 · (-0.5).
Step 3: Result is -1.
Answer: Lines are perpendicular.
A horizontal line is parallel to the x-axis and has a constant y-coordinate, resulting in a slope of 0. A vertical line is parallel to the y-axis and has a constant x-coordinate, which makes the denominator of the slope formula zero, rendering the slope undefined.
Step 1: The equation x = 5 represents a vertical line.
Step 2: For any two points (5, y₁) and (5, y₂), the change in x is 0.
Step 3: Division by zero is undefined.
Answer: Slope is undefined.