Question

Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

A.) f(x) = −16x^2
B.) f(x) = 16x^2
C.) f(x) = −1/16 x^2
D.) f(x) = 1/16 x^2

Answer 1

`\sqrt{(x_(0) - 0)^(2) + (y_(0) - (-4))^(2)} = |y_(0) - 4|`

`\sqrt{(x_(0) - 0)^(2) + (y_(0) + 4)^(2)} = |y_(0) - 4|`

`(x_(0) - 0)^(2) + (y_(0) + 4)^(2) = (y_(0) - 4)^(2)`

`(x_(0))^(2) + (y_(0) + 4)^(2) = (y_(0) - 4)^(2)`

`x_(0)^(2) + (y_(0)^(2) + 4y_(0) + 4y_(0) + 16) = y_(0)^(2) - 4y_(0) - 4y_(0) + 16`

`x_(0)^(2) + (y_(0)^(2) + 8y_(0) + 16) = y_(0)^(2) - 8y_(0) + 16`

`x_(0)^(2) + y_(0)^(2) + 8y_(0) + 16 = y_(0)^(2) - 8y_(0) + 16`

`x_(0)^(2) + 16y_(0) = 0`

`16y_(0) = -x_(0)^(2)`

`(16y_(0))/(16) = (-x_(0)^(2))/(16)`

`y_(0) = -(1)/(16)x_(0)^(2)`

`y = -(1)/(16)x^(2)`

The answer is C.