Coordinate Geometry and Straight Lines
This domain rewards careful substitution: distance, section and midpoint formulas, area of a triangle, collinearity, slope, and the standard forms of a straight line. Most errors are sign errors inside brackets.
Core rules
- Distance: √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Section formula for the point dividing AB in the ratio m : n internally, counted from A(x₁, y₁): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
- Area of a triangle: ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|; three points are collinear exactly when this area is 0.
- Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have m₁ = m₂, perpendicular lines have m₁m₂ = −1.
- Line forms: y = mx + c; y − y₁ = m(x − x₁); x/a + y/b = 1 for intercepts a and b.
Worked example
In what ratio does the point (−4, 6) divide the join of A(−6, 10) and B(3, −8)? Let the ratio be m : n measured from A. The x-coordinate gives (3m − 6n)/(m + n) = −4, so 3m − 6n = −4m − 4n, giving 7m = 2n and m : n = 2 : 7. Check with y: (2(−8) + 7(10))/9 = (−16 + 70)/9 = 54/9 = 6, as required.
Traps
- Pairing x₁ with n instead of m when writing the section formula.
- Reporting the ratio in the reverse order from the point named first in the question.
- Omitting the absolute value in the area formula and returning a negative area.
- Writing the perpendicular condition as m₁ = −m₂ rather than m₁m₂ = −1.
- Mixing up the x-intercept and y-intercept in x/a + y/b = 1.