Question

Derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7.

f(x) = −one twenty fourth(x − 1)2 − 5
f(x) = one twenty fourth(x − 1)2 − 5
f(x) = −one twenty fourth(x + 5)2 + 1
f(x) = one twenty fourth(x + 5)2 + 1

Answer 1

f(x) = −one twentyfourth (x + 5)2 + 1

Explanation:

Answer 2

`y = - (1)/(24) (x + 5) + 1`

Explanation

The directrix y=7, is above the y-value of the focus. The parabola must will open downwards.

Such parabola has equation of the form,

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

where (h,k) is the vertex.

The vertex is the midway from the focus to the directrix

The x-value of the vertex is x=-5 because it is on a vertical line that goes through (-5,-5).

The y-value of the vertex is

`y = ( 7 + - 5)/(2)`

`y = ( 2)/(2) = 1`

The equation of the parabola now becomes

`{(x + 5)}^(2) = - 4p(y - 1)`

p is the distance from the focus to the vertex which is p=|7-1|=6

Substitute the value of p to get:

`{(x + 5)}^(2) = - 4 * 6(y - 1)`

`{(x + 5)}^(2) = - 24(y - 1)`

We solve for y to get:

`y = - (1)/(24) (x + 5) + 1`