Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Coordinate Geometry – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Coordinate Geometry chapter of Mathematics. 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 between two points P(x₁, y₁) and Q(x₂, y₂): √((x₂ - x₁)² + (y₂ - y₁)²)
Calculates the straight-line distance between two points in a plane. x₁, y₁ are coordinates of point P; x₂, y₂ are coordinates of point Q. Essential for finding lengths in geometry problems.
Mid-point formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)
Finds the midpoint of a line segment joining two points P(x₁, y₁) and Q(x₂, y₂). Useful in dividing a line segment into two equal parts.
Section formula (internal division): ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))
Divides a line segment internally in the ratio m₁:m₂. m₁ and m₂ are the ratios; (x₁, y₁) and (x₂, y₂) are the endpoints. Key for problems involving ratio division.
Area of a triangle using coordinates: ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Calculates the area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃). Absolute value ensures the area is non-negative.
Slope of a line (m): (y₂ - y₁)/(x₂ - x₁)
Measures the steepness of a line joining two points (x₁, y₁) and (x₂, y₂). Positive slope indicates an upward trend; negative, downward.
Equation of a line: y - y₁ = m(x - x₁)
Finds the equation of a line with slope m passing through point (x₁, y₁). Fundamental for graphing linear equations.
Distance of a point P(x, y) from origin: √(x² + y²)
Special case of the distance formula where one point is the origin (0,0). Useful in problems involving central symmetry.
Centroid of a triangle: ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
The intersection point of the medians of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃). Represents the average position of the three vertices.
Collinearity condition: |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| = 0
Determines if three points are collinear (lie on the same straight line). Derived from the area of a triangle formula.
Section formula (external division): ((m₁x₂ - m₂x₁)/(m₁ - m₂), (m₁y₂ - m₂y₁)/(m₁ - m₂))
Divides a line segment externally in the ratio m₁:m₂. Useful for problems where the dividing point lies outside the segment.
Equations
Distance from point (x, y) to line Ax + By + C = 0: |Ax + By + C|/√(A² + B²)
Calculates the perpendicular distance from a point to a line. A, B, C are coefficients of the line equation; (x, y) is the point. Essential for shortest distance problems.
Equation of x-axis: y = 0
The x-axis is the horizontal axis where all points have a y-coordinate of 0. Fundamental for coordinate system basics.
Equation of y-axis: x = 0
The y-axis is the vertical axis where all points have an x-coordinate of 0. Fundamental for coordinate system basics.
Parallel lines condition: m₁ = m₂
Two lines are parallel if their slopes m₁ and m₂ are equal. Useful in identifying parallel lines without graphing.
Perpendicular lines condition: m₁ * m₂ = -1
Two lines are perpendicular if the product of their slopes m₁ and m₂ is -1. Key for problems involving right angles.
General equation of a line: Ax + By + C = 0
Represents any straight line in the plane. A, B, C are constants. Standard form for linear equations.
Slope-intercept form: y = mx + c
m is the slope; c is the y-intercept. Simplifies graphing linear equations by directly showing slope and y-intercept.
Intercept form of a line: x/a + y/b = 1
a and b are x and y intercepts respectively. Useful for quickly sketching lines when intercepts are known.
Point-slope form: y - y₁ = m(x - x₁)
Derived from the slope definition. Directly relates a point (x₁, y₁) and slope m to the line's equation.
Two-point form of a line: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)
Uses two points (x₁, y₁) and (x₂, y₂) to define a line. Alternative to point-slope form when two points are known.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
