Analytical Geometry (Two-Dimensional)
What the exam tests
Two-dimensional analytical geometry questions cover locus, straight lines, and circles: finding equations, slopes, intersection points, distances, centres and radii. The emphasis is on translating a geometric condition into an algebraic equation.
Core rules
- Slope m = (y₂ − y₁)/(x₂ − x₁); vertical lines have undefined slope.
- Line forms: y = mx + c; y − y₁ = m(x − x₁); x/a + y/b = 1; general ax + by + c = 0 with slope −a/b.
- Parallel: m₁ = m₂. Perpendicular: m₁m₂ = −1.
- Distance of (x₁, y₁) from ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²).
- Circle x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
Worked example
Find the equation of the line through (2, 3) and (4, 7). m = (7 − 3)/(4 − 2) = 4/2 = 2. Using point-slope form: y − 3 = 2(x − 2), so y = 2x − 1. Check: at x = 4, y = 7, so the second point satisfies it. The line perpendicular to this one through (2, 3) has slope −1/2 and equation y − 3 = −½(x − 2), i.e. x + 2y = 8.
For x² + y² − 4x + 6y − 12 = 0: g = −2, f = 3, c = −12; centre (2, −3), radius √(4 + 9 + 12) = 5.
Traps
- Sign errors when reading g and f; centre is (−g, −f), not (g, f).
- Forgetting the modulus in the distance formula, producing a negative distance.
- Using m₁ = m₂ for perpendicular lines.
- Failing to simplify the circle equation before identifying centre and radius.