Find the standard form of the parabola with the given characteristics.
focus: $( 8,13 )$ directrix: $y = - 1$
A) $( x - 8 ) ^ { 2 } = 28 ( y - 6 )$
B) $( x + 8 ) ^ { 2 } = 28 ( y + 6 )$
C) $( x - 6 ) ^ { 2 } = 28 ( y - 8 )$
D) $( x + 6 ) ^ { 2 } = 28 ( y + 8 )$
E) $( x + 8 ) ^ { 2 } = 7 ( y + 6 )$

Question

Find the standard form of the parabola with the given characteristics.
focus: $( 8,13 )$               directrix: $y = - 1$
A) $( x - 8 ) ^ { 2 } = 28 ( y - 6 )$
B) $( x + 8 ) ^ { 2 } = 28 ( y + 6 )$
C) $( x - 6 ) ^ { 2 } = 28 ( y - 8 )$
D) $( x + 6 ) ^ { 2 } = 28 ( y + 8 )$
E) $( x + 8 ) ^ { 2 } = 7 ( y + 6 )$

Quizplus · Accepted Answer

The standard form of a parabola is $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus or directrix. The focus is at $(8, 13)$ and the directrix is $y = -1$. The vertex lies midway between the focus and directrix, so its y-coordinate is $\frac{13 + (-1)}{2} = 6$. The x-coordinate of the vertex is the same as the focus, so $h = 8$. The distance $p$ is half the distance between the focus and directrix, which is $14 / 2 = 7$. Thus, $4p = 28$. The correct equation is $(x - 8)^2 = 28(y - 6)$.

Find the Standard Form of the Parabola with the Given (8,13)( 8,13 )(8,13)

Find the Standard Form of the Parabola with the Given $( 8,13 )$