Question

Which is the equation of a parabola with focus (0, 5) and directrix y= -5?

Answer 1

Answer:

Parabolic Equation:  20*y = $x^{2}$

or y = (1/20)*$x^{2}$

Step-by-step explanation:

general formula for parabolic conic section:

4c *(y - k) = (x - h)^2

y = (1/4c)*(x - h)^2  + k ;   a = 1 / 4c  

with focus at   (h, k + c)

directrix:  y = k - c

focus: (0, 5) = (h, k + c)

directrix  y = -5 = k - c

so  h = 0,  5 = k + c

-5 = k - c

solve the equation   5 = k + c  and  -5 = k - c

We get:   0 = 2k ,   k = 0

so c = 5

4*c (y - k) = (x - h)^2

4*5*(y - 0) = (x - 0)^2

20*y = $x^{2}$

or y = (1/20)*$x^{2}$

Answer 2

Answer:

Step-by-step explanation:

Let P(x,y) be any point on the parabola.

focus S (0,5)

Directrix y=-5

or y+5=0

let M be foot of perpendicular from P(x,y) on directrix.

SP=$SP=\sqrt{(x-0)^2+(y-5)^2}\\ PM=\frac{y+5}{\sqrt{1} } =y+5$

SP=PM

SP²=PM²

(x-0)²+(y-5)²=(y+5)²

x²=(y+5)²-(y-5)²

or x²=y²+10y+25-(y²-10y+25)

x²=y²+10y+25-y²+10y-25

x²=20y