Summary
Coordinate geometry involves understanding the properties and equations of lines and circles on a Cartesian plane.
- Length of a Line Segment — the distance between two points. Example: The distance between points A(5, 2) and B(1, -1) is calculated using the distance formula.
- Midpoint of a Line Segment — the point that divides a line segment into two equal parts. Example: The midpoint of line AB with endpoints A(5, 2) and B(1, -1) is calculated using the midpoint formula.
- Parallel Lines — lines with equal gradients. Example: Lines AB and PQ are parallel if their gradients are equal.
- Perpendicular Lines — lines whose gradients multiply to -1. Example: Lines AB and BC are perpendicular if the product of their gradients is -1.
- Equation of a Straight Line — can be expressed in various forms such as y = mx + c. Example: The line passing through (x1, y1) with gradient m is given by y - y1 = m(x - x1).
- Equation of a Circle — represents a circle with a given center and radius. Example: The equation (x-a)² + (y-b)² = r² describes a circle with center (a, b) and radius r.
Exam Tips
Key Definitions to Remember
- Length of a line segment
- Midpoint of a line segment
- Gradient of a line
- Equation of a circle
Common Confusions
- Mixing up the formulas for parallel and perpendicular lines
- Forgetting to square the radius in the circle equation
Typical Exam Questions
- How do you find the distance between two points? Use the distance formula: √((x2-x1)² + (y2-y1)²)
- What is the equation of a line parallel to y = 3x + 2? Any line with gradient 3, e.g., y = 3x + c
- How do you find the center and radius from the circle equation x² + y² + 2gx + 2fy + c = 0? Center is (-g, -f) and radius is √(g² + f² - c)
What Examiners Usually Test
- Ability to derive and use the equation of a line
- Understanding of the relationship between lines and circles
- Solving problems involving intersections of lines and circles