Question

Derive the equation of the parabola with a focus at (-5,5) and a directix of y = -1

Answer 1

D

Explanation:

From any point (x, y) on the parabola the focus and directrix are equidistant

Using the distance formula

`√((x+5)^2+(y-5)^2)` = | y + 1 |

Squaring both sides

(x + 5)² + (y - 5)² = (y + 1)^2 , that is

(y + 1)² = (x + 5)² + (y - 5)² ← subtract (y - 5)² from both sides

(y + 1)² - (y - 5)² = (x + 5)² ← expand left side and simplify

y² + 2y + 1 - y² + 10y - 25 = (x + 5)²

12y - 24 = (x + 5)² ← factor left side

12(y - 2) = (x + 5)² ← divide both sides by 12

y - 2 =
`(1)/(12)` (x + 5)² ← add 2 to both sides

y =
`(1)/(12)` (x + 5)² + 2

or

f(x) =
`(1)/(12)` (x + 5)² + 2 → D