Coordinate Geometry Quiz

Coordinate Geometry

• Coordinate geometry deals with geometric figures in a coordinate plane (typically the Cartesian plane).
• Points are represented by ordered pairs (x, y) where x is the horizontal coordinate (abscissa) and y is the vertical coordinate (ordinate).
• The distance between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula: √((x₂ - x₁)² + (y₂ - y₁)²).
• The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).
• The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by (y₂ - y₁)/(x₂ - x₁).
• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• Equations of lines can be written in various forms, including point-slope form, slope-intercept form (y = mx + b), and standard form (Ax + By = C).
• The equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².
• The equation of a parabola with vertex (h, k) opens upward is (y - k) = a(x - h)² + c.
• Different conic sections (circles, ellipses, parabolas, hyperbolas) have specific equations that can be graphed and analyzed using coordinate geometry principles.

Simultaneous Equations

• Simultaneous equations are a set of two or more equations with the same variables.
• The goal is to find the values of the variables that satisfy all the equations simultaneously.
• Methods for solving simultaneous equations include:
  • Substitution: Solve one equation for one variable, and substitute the expression into the other equation.
  • Elimination: Add or subtract the equations to eliminate one variable.
  • Graphical Method: Graph the equations and find the point(s) of intersection.
• Linear simultaneous equations (with variables raised to the power of 1) often have a single unique solution, no solution (parallel lines), or infinitely many solutions (the same line).
• Non-linear simultaneous equations (variables raised to a power other than 1) can have more than one solution.
• The solution to a system of equations represents a point that lies on all lines in the system.

Sequences

• A sequence is an ordered list of numbers.
• The numbers in a sequence are called terms.
• Sequences can be arithmetic, geometric, or neither.
• Arithmetic sequences have a constant difference between consecutive terms (common difference).
• Geometric sequences have a constant ratio between consecutive terms (common ratio).
• The nth term of an arithmetic sequence can be found using the formula: an = a₁ + (n-1)d, where a₁ is the first term and d is the common difference.
• The nth term of a geometric sequence can be found using the formula: an = a₁ * r^(n-1), where a₁ is the first term and r is the common ratio.
• Series are the sum of the terms in a sequence.
• The sum of an arithmetic series can be calculated using the formula: Sn = n/2 (2a₁ + (n-1)d).
• The sum of a finite geometric series can be calculated using the formula: Sn = a₁ (1 - r^n) / (1 - r), where r ≠ 1.
• Recurrence relations define terms in a sequence based on previous terms in the sequence.
• Special types of sequences include Fibonacci sequences, where each term is the sum of the two preceding terms.
• Understanding sequences is valuable in many areas of mathematics and beyond, helping model patterns and behavior in various applications.