Geometric loci and conic sections: definitions, elements and constructions
This block tests whether you can work from a definition instead of a formula. The exam gives data - foci, axis, directrix, a point - and expects the curve to be generated by the locus property, with the elements named correctly.
Loci
The locus is the set of points satisfying one condition. The examinable ones are: perpendicular bisector (equidistant from two points), angle bisector (equidistant from two lines), circle (fixed distance from a point), parallels at a given distance from a line, and the arc capable - the locus of points seeing segment AB under an angle α, which is two arcs of radius R = AB/(2·sin α) drawn from a centre lying on the perpendicular bisector.
The conics as loci
- Ellipse: PF₁ + PF₂ = 2a, with a² = b² + c² and eccentricity e = c/a < 1. The foci lie inside, at distance c from the centre.
- Hyperbola: |PF₁ − PF₂| = 2a, with c² = a² + b² and e = c/a > 1; the asymptotes pass through the centre and the rectangle 2a × 2b.
- Parabola: distance to the focus equals distance to the directrix, e = 1; the vertex is midway between focus and directrix, and the parameter p = 2·VF. The circle is the ellipse with c = 0, hence e = 0.
Standard constructions
Ellipse by points: take r₁ from the focus, then r₂ = 2a − r₁, and intersect the two arcs; repeat symmetrically. Ellipse by concentric circles (affinity with radii a and b), and by the parallelogram method for a conjugate-diameter ellipse. Parabola from focus and directrix: a parallel to the directrix at distance d meets an arc of radius d centred on the focus. Hyperbola from foci and 2a by the same difference-of-distances arcs, or from the asymptotes and one point.
Worked example
An ellipse has focal distance 2c = 40 mm and major axis 2a = 50 mm. Then c = 20, a = 25, b = √(625 − 400) = 15 mm and e = 20/25 = 0.8. For a point with r₁ = 30 mm, r₂ = 50 − 30 = 20 mm, so the point is the intersection of the circle centred on F₁ with radius 30 and the circle centred on F₂ with radius 20.
Traps
Confusing a with 2a; using c² = a² − b² in the hyperbola, where c² = a² + b²; placing the directrix through the focus; calling the focal distance the major axis; forgetting that the ellipse constructions are symmetric, so each point gives four.