Question

Derive an analytic equation for a parabola whose focus is (0,4) and directrix is the x-axis. Explain how you got your answer.

Answer 1

y =x²÷8 + 2

Explanation:

general equation for a parabola is: ×: ( ×-h)² = +4*p*(y-k)

where the vertex v ( h, k ) is the point ( 0, 2) and the directrix is a line over the x-axis. Vertex is half way between x-axis and focus, the distance between vertex and x-axis is p. Sign + in the right hand side of the equation means the parabola open upwards

So (×-0)² = +4*2 (y-2) ⇒ x²= 8*(y-2) ⇒ x² = 8*y -16

x² = 8y - 16 ⇒ 8y = ײ + 16 ⇒ y =x²÷8 + 2