Circles and Conic Sections
Questions cover the equation of a circle (centre, radius, tangency) and the standard forms of parabola, ellipse and hyperbola with their foci, directrices and eccentricities.
Circle
General form x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c), real only when g² + f² − c > 0. Standard form: (x−h)² + (y−k)² = r². A circle with (x₁,y₁) and (x₂,y₂) as diameter ends: (x−x₁)(x−x₂) + (y−y₁)(y−y₂) = 0.
Conics
- Parabola y² = 4ax: focus (a,0), directrix x = −a, latus rectum 4a, eccentricity 1.
- Ellipse x²/a² + y²/b² = 1 (a > b): foci (±ae, 0), e = √(1 − b²/a²) < 1.
- Hyperbola x²/a² − y²/b² = 1: foci (±ae, 0), e = √(1 + b²/a²) > 1, asymptotes y = ±(b/a)x.
Worked example
For x² + y² − 4x + 6y − 12 = 0, compare with the general form: 2g = −4 so g = −2; 2f = 6 so f = 3; c = −12. Centre = (2, −3) and radius = √(4 + 9 + 12) = √25 = 5.
Traps that cost marks
- Writing the centre as (g, f) instead of (−g, −f).
- Computing the radius as √(g² + f² + c); c is subtracted.
- Swapping the ellipse and hyperbola eccentricity formulae (e < 1 for ellipse, e > 1 for hyperbola).
- Forgetting the factor 2 when extracting g and f from 2gx and 2fy.