Straight Lines

The slope (or gradient) of a non-vertical line is a measure of its steepness. It is denoted by 'm'.\n\n1. **Angle of Inclination (θ)**: The angle θ (0 ≤ θ < π) made by a line with the positive direction of the x-axis, measured anti-clockwise, is called its inclination.\n2. **Slope (m)**: If θ is the inclination of a line, then its slope m is given by m = tan θ. If θ = π/2 (i.e., the line is vertical), the slope is undefined.\n3. **Slope of a line passing through two points (x₁, y₁) and (x₂, y₂)**: If x₁ ≠ x₂, the slope m is given by the ratio of the difference in y-coordinates to the difference in x-coordinates.\n4. **Condition for Parallel Lines**: Two non-vertical lines are parallel if and only if their slopes are equal (m₁ = m₂).\n5. **Condition for Perpendicular Lines**: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 (m₁m₂ = -1). If one line is vertical, the other must be horizontal.\n6. **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.

The equation of a straight line can be expressed in various forms depending on the given information.\n\n1. **Horizontal and Vertical Lines**:\n * Equation of a horizontal line (parallel to x-axis) at a distance 'k' from the x-axis is y = k.\n * Equation of a vertical line (parallel to y-axis) at a distance 'k' from the y-axis is x = k.\n2. **Point-Slope Form**: The equation of a line passing through a point (x₁, y₁) with slope 'm' is given by:\n3. **Two-Point Form**: The equation of a line passing through two distinct points (x₁, y₁) and (x₂, y₂) is given by:\n4. **Slope-Intercept Form**: The equation of a line with slope 'm' and y-intercept 'c' is given by:\n5. **Intercept Form**: The equation of a line making x-intercept 'a' and y-intercept 'b' is given by:\n6. **Normal Form**: The equation of a line where 'p' is the length of the perpendicular from the origin to the line, and 'α' is the angle which the normal (perpendicular) makes with the positive direction of the x-axis, is given by:\n7. **General Equation of a Line**: Any equation of the form Ax + By + C = 0, where A, B, C are real numbers and A and B are not both zero, represents a straight line. From this form:\n * Slope m = -A/B (if B ≠ 0)\n * x-intercept = -C/A (if A ≠ 0)\n * y-intercept = -C/B (if B ≠ 0)