Analytical Geometry
Analytical Geometry turns points, lines and conic sections into coordinates and equations. The exam tests distance and section formulas, slopes and line equations, circle equations, and the standard forms of parabola, ellipse and hyperbola.
Core rules
- Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
- 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 ax + by + c = 0, slope = −a/b.
- Circle: (x−h)² + (y−k)² = r² has centre (h,k), radius r. In x²+y²+2gx+2fy+c = 0, centre is (−g,−f) and radius is √(g²+f²−c).
- Conics: y² = 4ax has focus (a,0) and directrix x = −a. For x²/a² + y²/b² = 1 with a > b, e = √(1−b²/a²). For x²/a² − y²/b² = 1, e = √(1+b²/a²).
Worked example
Find the circle with centre (2,−3) and radius 5. Standard form: (x−2)² + (y+3)² = 25. Expand: x²−4x+4 + y²+6y+9 = 25, so x²+y²−4x+6y−12 = 0. Check the centre from the general form: (−g,−f) = (2,−3), correct.
Traps
Reading the centre as (g,f) instead of (−g,−f); forgetting that a circle needs equal x² and y² coefficients and no xy term; using the reciprocal instead of the negative reciprocal for a perpendicular slope; and confusing r² with r in the radius formula.