Coordinates: Distance, Midpoint, Slope and Collinearity
What the exam tests
Coordinate geometry questions test the distance formula, midpoint, section formula, slope, parallel and perpendicular lines, and collinearity of three points. Kerala SSLC problems often combine two of these ideas in one question.
Core rules
- Distance between (x₁,y₁) and (x₂,y₂): √[(x₂−x₁)²+(y₂−y₁)²].
- Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2).
- Section formula for dividing AB in ratio m:n: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)).
- Slope: m=(y₂−y₁)/(x₂−x₁).
- Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
- Three points are collinear if the slope between each pair is the same.
Worked example
For A(1,2) and B(7,10), distance AB=√[(7−1)²+(10−2)²]=√(36+64)=10. Midpoint=(4,6). Slope AB=(10−2)/(7−1)=8/6=4/3. A line through A with slope 4/3 has equation y−2=(4/3)(x−1), which simplifies to 3y−4x−2=0.
Traps
Subtract coordinates in the same order in both brackets. Distance is non-negative; a negative answer signals an arithmetic slip. The midpoint is an average, not a difference. In the section formula, match the ratio to the correct endpoint. A vertical line has undefined slope, not zero. For perpendicular lines, use the negative reciprocal, not just the reciprocal.