©G Dear2008 – Not to be sold/Free to use Mathematic (Preliminary) Locus and the Parabola Harder Parabolas Stage 6 - Year 11 Press Ctrl-A ©G Dear2008 – Not to be sold/Free to use

Parabola as a locus Vertex is at (h, k) Focus is at (h, k+a) The parabola, vertex (h, k) and focal length a, has the equation (x – h)2 = 4a(y – k). Vertex is at (h, k) x = h (h, k + a) Focus is at (h, k+a) a Focal Length a (h, k) Axis is x = h Directrix is y = k-a y = k -a

Parabola as a locus Find the equation of the parabola with focus (2,3) and with directrix y =-7. a =(3 + 7)/2 = 5 Vertex = (2,3-5) = (2,-2) = (h, k) (2, 3) a=5 (x – h)2 = 4a(y – k) (x – 2)2 = 4(5)(y – -2) (2, -2) x2 – 4x + 4 = 20 (y + 2) x2 – 4x + 4 = 20y + 40 y -7 x2–4x-20y-36=0