Question

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

Answer 1

$\sqrt{(x_{0} - 0)^{2} + (y_{0} - (-4))^{2}} = |y_{0} - 4|$
$\sqrt{(x_{0} - 0)^{2} + (y_{0} + 4)^{2}} = |y_{0} - 4|$
$(x_{0} - 0)^{2} + (y_{0} + 4)^{2} = (y_{0} - 4)^{2}$
$(x_{0})^{2} + (y_{0} + 4)^{2} = (y_{0} - 4)^{2}$
$x_{0}^{2} + (y_{0}^{2} + 4y_{0} + 4y_{0} + 16) = y_{0}^{2} - 4y_{0} - 4y_{0} + 16$
$x_{0}^{2} + (y_{0}^{2} + 8y_{0} + 16) = y_{0}^{2} - 8y_{0} + 16$
$x_{0}^{2} + y_{0}^{2} + 8y_{0} + 16 = y_{0}^{2} - 8y_{0} + 16$
$x_{0}^{2} + 16y_{0} = 0$
$16y_{0} = -x_{0}^{2}$
$\frac{16y_{0}}{16} = \frac{-x_{0}^{2}}{16}$
$y_{0} = -\frac{1}{16}x_{0}^{2}$
$y = -\frac{1}{16}x^{2}$

The answer is C.