Coordinate Geometry

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The slope (m) of a line is tan θ, where θ is the inclination angle with the positive x-axis. For a line passing through points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁)/(x₂ - x₁). The slope represents the …

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Parallel lines: m₁ = m₂ (slopes are equal). Perpendicular lines: m₁ × m₂ = -1 (product of slopes equals -1). Note: This applies only to non-vertical lines.

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Point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a known point on the line. Use this form when you know the slope and one point on the line.

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Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept (the point where the line crosses the y-axis). This is useful when you know the slope and y-intercept.

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Intercept form: x/a + y/b = 1, where 'a' is the x-intercept and 'b' is the y-intercept. Use this when you know both intercepts. The line passes through points (a, 0) and (0, b).

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Normal form: x cos α + y sin α = p, where p is the perpendicular distance from origin to the line and α is the angle the perpendicular makes with the positive x-axis (0° ≤ α < 360°).

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General form: Ax + By + C = 0, where A² + B² ≠ 0 (A and B cannot both be zero simultaneously). This represents a straight line for any values of A, B, C satisfying this condition.

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Distance from point (x₁, y₁) to line Ax + By + C = 0 is: d = |Ax₁ + By₁ + C|/√(A² + B²). The absolute value ensures distance is always positive.