Coordinate Geometry and Straight Lines
Coordinate Geometry and Straight Lines tests algebraic descriptions of points and lines: distance, section formula, area, slope, and the standard forms of a line. Accuracy with signs matters more than memorising many forms.
Core rules
- Distance between (x₁,y₁) and (x₂,y₂): √[(x₂−x₁)²+(y₂−y₁)²].
- Section formula for ratio m:n: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); midpoint is m=n.
- Area of triangle: ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.
- Slope m=(y₂−y₁)/(x₂−x₁); parallel lines have equal slopes, perpendicular lines have product −1.
- Forms: y−y₁=m(x−x₁); y=mx+c; x/a+y/b=1; general ax+by+c=0 has slope −a/b.
Worked example
Find the line through (2,−1) perpendicular to 3x+4y=7. The given slope is −3/4, so required slope is 4/3. y+1=(4/3)(x−2), i.e. 4x−3y−11=0.
Traps
The perpendicular slope is the negative reciprocal, not the reciprocal. A vertical line has undefined slope; write x=k, not y=mx+c. In the area formula, keep the absolute value and consistent vertex order. In section formula, do not swap the ratio’s near and far points.