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

Answer:

y =x²÷8  + 2

Step-by-step 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