Coordinate Geometry – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Coordinate Geometry chapter of Mathematic. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Distance Formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
d is the distance between points (x₁, y₁) and (x₂, y₂). This formula is derived from the Pythagorean theorem to calculate the distance in a Cartesian plane.
Midpoint Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
M is the midpoint between points (x₁, y₁) and (x₂, y₂). It finds the center point of a line segment, useful in geometry and various applications.
Section Formula: P(x, y) = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))
P is the point dividing the line joining (x₁, y₁) and (x₂, y₂) in the ratio m:n. This is useful for partitioning segments in given ratios.
Slope of a Line: m = (y₂ - y₁) / (x₂ - x₁)
m is the slope indicating the steepness of the line. It's crucial in defining the angle of a line and analyzing linear relationships.
Equation of a Line (Slope-Intercept Form): y = mx + c
m is the slope, and c is the y-intercept. This form is key to quickly graphing linear equations.
General Form of a Line: Ax + By + C = 0
A, B, and C are constants. This form is a versatile way to represent linear equations and examine their geometric properties.
Distance of a Point from a Line: d = |Ax₀ + By₀ + C| / √(A² + B²)
This formula calculates the perpendicular distance d from the point (x₀, y₀) to the line Ax + By + C = 0, useful in optimization problems.
Area of Triangle: Area = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
This formula calculates the area based on vertex coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃). It's useful in both geometry and real-world applications.
Area of Quadrilateral: Area = 1/2 |x₁(y₂ - y₄) + x₂(y₃ - y₁) + x₃(y₁ - y₄) + x₄(y₁ - y₂)|
This finds the area by utilizing the vertices coordinates of the quadrilateral. Important for solving geometric problems.
Equidistant Point: |PA| = |PB| yields (x - x₁)² + (y - y₁)² = (x - x₂)² + (y - y₂)²
This defines the locus of points equidistant from two given points A and B. It's key to understanding geometric relationships.
Equations
Linear equation: ax + by + c = 0
This equation represents a straight line and defines relationships in coordinate geometry.
Slope formula from two points: m = (y₂ - y₁) / (x₂ - x₁)
Calculating the slope m provides insights into the angle of inclination of the line segment between points (x₁, y₁) and (x₂, y₂).
Point of intersection (two lines): Solve {y = mx + c₁; y = nx + c₂}
Finding the intersection helps in solving real-world problems where lines converge or meet.
Perpendicular lines: m₁ * m₂ = -1
If two lines are perpendicular, the product of their slopes is -1, indicating the lines meet at right angles.
Distance from origin to point (x, y): d = √(x² + y²)
This formula calculates the straight-line distance from the origin to a point in a Cartesian plane.
Equation of a vertical line: x = k
Describes a line parallel to the y-axis at x = k; useful for identifying specific vertical positions on the coordinate plane.
Equation of a horizontal line: y = k
Describes a line parallel to the x-axis at y = k; useful in defining horizontal boundaries on graphs.
Circle equation: (x - h)² + (y - k)² = r²
Defines a circle with center (h, k) and radius r, foundational for many geometry applications.
Equation of a parabola: y = ax² + bx + c
This represents a quadratic function and demonstrates the application of coordinate geometry in graphing parabolas.
Equation of a hyperbola: (x²/a²) - (y²/b²) = 1
Provides the standard form of a hyperbola, which is another essential conic section in coordinate geometry.
