Coordinate Geometry and Conic Sections
The Coordinate Geometry and Conic Sections domain covers the straight line, the circle and the three standard conics in Cartesian and parametric form. The exam tests your ability to extract geometric data - centre, radius, focus, directrix, eccentricity, asymptotes - from an equation, and to impose conditions such as tangency or perpendicularity.
Core rules
- Line through (x1, y1) with gradient m: y - y1 = m(x - x1); ax + by + c = 0 has gradient -a/b. Perpendicular distance from (x1, y1) is |ax1 + by1 + c|/sqrt(a^2 + b^2).
- Circle x^2 + y^2 + 2gx + 2fy + c = 0 has centre (-g, -f) and radius sqrt(g^2 + f^2 - c).
- Parabola y^2 = 4ax: focus (a, 0), directrix x = -a, latus rectum of length 4a, parametric point (at^2, 2at), tangent ty = x + at^2.
- Ellipse x^2/a^2 + y^2/b^2 = 1: eccentricity e = sqrt(1 - b^2/a^2), foci (±ae, 0), parametric point (a cos θ, b sin θ).
- Hyperbola x^2/a^2 - y^2/b^2 = 1: e = sqrt(1 + b^2/a^2), asymptotes y = ±(b/a)x; for the rectangular hyperbola xy = c^2 the asymptotes are the axes.
Worked example
For y^2 = 12x, comparing with y^2 = 4ax gives a = 3. The focus is (3, 0), the directrix is x = -3 and the latus rectum has length 4a = 12. A parametric point is (3t^2, 6t); the tangent there follows from yy1 = 2a(x + x1), giving 6ty = 6(x + 3t^2), that is ty = x + 3t^2.
Traps
- Misreading the sign of a: y^2 = -12x opens to the left with focus (-3, 0).
- Completing the square wrongly for a circle, especially the signs of g and f in the centre.
- Using e = sqrt(1 - b^2/a^2) for a hyperbola, where the plus sign is required.
- Forgetting that a general second-degree equation gives a conic only when the discriminant condition on its quadratic terms holds.