Conic sections and the parabola
Recognise the parabola as the set of points equally far from a focus and a directrix, and read its equation with the vertex at the origin and at any point.
- Iran: grade 12
- Afghanistan: grade 11
- Syria: grade 10
- Palestine: grade 11
Guide
Cut a cone with a plane and the cut is a curve called a conic section. The parabola, ellipse and hyperbola are conic sections, and each can be defined by distances from fixed points or a fixed line.
A parabola is the set of points equally far from a fixed point F (the focus) and a fixed line D (the directrix): MF = MK. With the vertex at the origin and focus F(p,0), the directrix is x = −p and the equation is y2 = 4px.
With vertex S(h,k) and the parabola opening up or down, the equation is (x − h)2 = 4p(y − k); the focus is F(h,k + p) and the directrix y = k − p. If p is positive the parabola opens upwards.