t6onNeliyaSY
contestada

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

Respuesta :

Panoyin
[tex]\sqrt{(x_{0} - 0)^{2} + (y_{0} - (-4))^{2}} = |y_{0} - 4|[/tex]
[tex]\sqrt{(x_{0} - 0)^{2} + (y_{0} + 4)^{2}} = |y_{0} - 4|[/tex]
[tex](x_{0} - 0)^{2} + (y_{0} + 4)^{2} = (y_{0} - 4)^{2}[/tex]
[tex](x_{0})^{2} + (y_{0} + 4)^{2} = (y_{0} - 4)^{2}[/tex]
[tex]x_{0}^{2} + (y_{0}^{2} + 4y_{0} + 4y_{0} + 16) = y_{0}^{2} - 4y_{0} - 4y_{0} + 16[/tex]
[tex]x_{0}^{2} + (y_{0}^{2} + 8y_{0} + 16) = y_{0}^{2} - 8y_{0} + 16[/tex]
[tex]x_{0}^{2} + y_{0}^{2} + 8y_{0} + 16 = y_{0}^{2} - 8y_{0} + 16[/tex]
[tex]x_{0}^{2} + 16y_{0} = 0[/tex]
[tex]16y_{0} = -x_{0}^{2}[/tex]
[tex]\frac{16y_{0}}{16} = \frac{-x_{0}^{2}}{16}[/tex]
[tex]y_{0} = -\frac{1}{16}x_{0}^{2}[/tex]
[tex]y = -\frac{1}{16}x^{2}[/tex]

The answer is C.