Analytical Geometry
Analytical geometry in the IPE papers means the circle, the conic sections (parabola, ellipse, hyperbola) and polar coordinates. Expect questions on centre and radius, tangents and normals in Cartesian and parametric form, chords of contact, families of circles, eccentricity, asymptotes and the conversion between polar and Cartesian coordinates.
Core rules and concepts
- Circle x² + y² + 2gx + 2fy + c = 0 has centre (−g, −f) and radius √(g² + f² − c).
- Length of the tangent from P(x₁, y₁) is √S₁, where S₁ is the left side evaluated at P. The chord of contact is T = 0.
- Parabola y² = 4ax: focus (a, 0), directrix x = −a, latus rectum 4a, parametric point (at², 2at), tangent ty = x + at².
- Ellipse x²/a² + y²/b² = 1: e = √(1 − b²/a²), foci (±ae, 0), sum of focal distances 2a.
- Hyperbola x²/a² − y²/b² = 1: e = √(1 + b²/a²), asymptotes x/a ± y/b = 0.
- Polar: x = r cos θ, y = r sin θ, with r allowed to be negative.
Worked example
Find the length of the tangent from (5, 4) to x² + y² − 4x − 2y − 11 = 0. Substitute: S₁ = 25 + 16 − 20 − 8 − 11 = 2. Tangent length = √S₁ = √2.
Traps that cost marks
- Sign errors on g and f; remember the centre is (−g, −f), not (g, f).
- Ignoring the condition g² + f² − c > 0 for a real circle.
- Using the ellipse eccentricity formula for a hyperbola.
- For the parabola y = mx + c, the tangency condition is c = a/m, not c = am.
- Dropping the modulus when a polar curve is described.