Question

Write the equation in vertex form for the parabola with focus (0,5) and directrix y=

Answer 1

Answer: $x^{2} = 20y$

Step-by-step explanation:

The directrix given is vertical , so we will use the formula :

$(x-h)^{2}=4p(y-k)$

P is the distance between the focus , that is 5 - 0 = 5

Therefore : p = 5

(h,k) is the mid point between the focus and the directrix , that is

(h,k) = $(\frac{x_{1}+x_{2} }{2},\frac{y_{2}+y_{1}}{2})$ = $(\frac{0+0}{2} , \frac{5-5}{2})$ = $(0,0)$

Therefore:

h =0

k = 0

substituting into the formula : we have

$(x-h)^{2}=4p(y-k)$

$(x-0)^{2}$ = 4(5)($y-0)$

$x^{2} = 20y$

Therefore : the equation in vertex form is $x^{2} = 20y$