Tangencies and technical curves (conics and their applications)
Tangency problems (line–circle, circle–circle, and the Apolonio cases) plus conic construction and tangents form a block of the Madrid paper. Every tangency answer must show the tangency point, because the point is what proves the contact: it lies on the line of centres or on the perpendicular from the centre to the line.
Core rules
- Circle tangent to two lines: its centre lies on the bisector of the angle.
- Tangent to a line at a given point: the centre lies on the perpendicular at that point.
- Circle tangent to another circle: distance between centres equals R + r (external tangency) or R − r (internal tangency), and the contact point lies on that line of centres.
- Difficult cases (tangent to two elements passing through an external point) are solved by homothety, by the radical axis, or by inversion.
- Ellipse: PF₁ + PF₂ = 2a, with a² = b² + c² and e = c/a < 1.
- Parabola: distance to the focus equals distance to the directrix, e = 1.
- Hyperbola: |PF₁ − PF₂| = 2a, e = c/a > 1, asymptotes through the centre.
- Tangents: at the ellipse the tangent bisects the exterior angle of triangle F₁PF₂ and the normal bisects the interior one; on the parabola the tangent bisects the angle between the focal radius and the line parallel to the axis.
Worked example
Ellipse with major axis 2a = 100 mm and focal distance 2c = 60 mm: a = 50, c = 30, b = √(50² − 30²) = 40 mm, so the minor axis is 80 mm and e = 0.6. For a point P with PF₁ = 35 mm, PF₂ must be 100 − 35 = 65 mm. Strike arcs of 35 and 65 mm from the foci; their intersection is P, and the tangent at P is the bisector of the exterior angle.
Traps that cost marks
Not marking the tangency point; choosing external tangency when the required circle must enclose the given one; writing e = a/c; confusing tangent and normal in the ellipse; forgetting that a plane section of a cone gives a parabola only when the plane is parallel to one generatrix; drawing the hyperbola through its asymptotes instead of asymptotic to them.