Summary
Coordinate geometry in the (x,y) plane involves understanding the equations and properties of lines and circles. It includes determining equations of straight lines, interpreting different forms of line equations, and understanding circle equations.
- Equation of a Straight Line — A mathematical expression representing a line in the plane. Example: y = mx + c, where m is the gradient and c is the y-intercept.
- Gradient (m) — The slope of a line, showing how steep it is. Example: For line AB with points A(x₁, y₁) and B(x₂, y₂), m = (y₂ - y₁) / (x₂ - x₁).
- Midpoint — The point exactly halfway between two points on a line segment. Example: Midpoint of AB is ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Equation of a Circle — Represents a circle in the plane with a given center and radius. Example: (x-a)² + (y-b)² = r², where (a, b) is the center and r is the radius.
- Parallel Lines — Lines in the same plane that never intersect. Example: Lines with equal gradients.
- Perpendicular Lines — Lines that intersect at a right angle. Example: Product of their gradients is -1.
Exam Tips
Key Definitions to Remember
- Equation of a straight line: y = mx + c
- Gradient of a line: (y₂ - y₁) / (x₂ - x₁)
- Equation of a circle: (x-a)² + (y-b)² = r²
Common Confusions
- Confusing the gradient formula with the midpoint formula
- Mixing up the forms of line equations
Typical Exam Questions
- What is the equation of a line passing through (3, 4) with a gradient of 2? y - 4 = 2(x - 3)
- How do you find the midpoint of a line segment with endpoints (1, 2) and (3, 4)? Midpoint is (2, 3)
- What is the radius of a circle with equation x² + y² - 4x - 6y + 9 = 0? Radius is 2
What Examiners Usually Test
- Ability to derive and use line equations
- Understanding of circle equations and their properties
- Solving problems involving intersections of lines and circles