Question

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

Answer 1

Parabolic Equation: 20*y =
`x^(2)`

or y = (1/20)*
`x^(2)`

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

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=√((x-0)^2+(y-5)^2)\\ PM=(y+5)/(√(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