🎯 Coordinate Geometry Problem Types
📊 Problem Pattern Catalog
Type 1: Distance and Midpoint
Pattern: Find distance between two points or midpoint of a segment Key Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Worked Example:
Find the distance between $A(3,4)$ and $B(-1,2)$.
Solution: $d = \sqrt{(-1-3)^2 + (2-4)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$
Type 2: Slope and Linear Equations
Pattern: Find slope, write equation of line, or find intersection Key Formula: $m = \frac{y_2-y_1}{x_2-x_1}$, $y - y_1 = m(x - x_1)$
Worked Example:
Write the equation of the line through $(2,5)$ and $(4,9)$.
Solution: First find slope: $m = \frac{9-5}{4-2} = \frac{4}{2} = 2$
Using point-slope form: $y - 5 = 2(x - 2)$
Simplifying: $y = 2x + 1$
Type 3: Parallel and Perpendicular Lines
Pattern: Find equation of parallel/perpendicular line through a point Key Concept: Parallel = same slope, Perpendicular = negative reciprocal
Worked Example:
Find the equation of the line perpendicular to $y = 3x - 2$ through $(1,4)$.
Solution: Original slope: $m_1 = 3$ Perpendicular slope: $m_2 = -\frac{1}{3}$
Equation: $y - 4 = -\frac{1}{3}(x - 1)$ Simplifying: $y = -\frac{1}{3}x + \frac{13}{3}$
Type 4: Circles
Pattern: Find center, radius, or equation of circle Key Formula: $(x-h)^2 + (y-k)^2 = r^2$
Worked Example:
Find the center and radius of $x^2 + y^2 - 4x + 6y - 3 = 0$.
Solution: Complete the square: $(x^2 - 4x) + (y^2 + 6y) = 3$ $(x^2 - 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9$ $(x-2)^2 + (y+3)^2 = 16$
Center: $(2,-3)$, Radius: $4$
Type 5: Transformations
Pattern: Apply translation, reflection, or rotation to points Key Concept: Know how each transformation affects coordinates
Worked Example:
Reflect the point $(3,5)$ over the x-axis, then translate 2 units right.
Solution: Reflection over x-axis: $(3,5) \to (3,-5)$ Translation 2 units right: $(3,-5) \to (5,-5)$
🔍 Problem-Solving Strategy
- Identify the type of coordinate geometry problem
- Write down relevant formulas for that type
- Substitute given values carefully
- Simplify step by step to avoid errors
- Check your answer by plugging back in
⚠️ Common Mistakes
- Sign errors in distance formula
- Mixing up rise and run in slope
- Forgetting to complete the square for circles
- Wrong direction in transformations
- Arithmetic errors in midpoint calculations