Find the standard form of the parabola with the given characteristics.
focus: $( 8,13 ) \quad$ directrix: $y = - 1$
A) $\quad ( x - 8 ) ^ { 2 } = 28 ( y - 6 )$
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

Quizplus · Accepted Answer

The focus of the parabola is at (8, 13) and the directrix is the line $y = -1$. The vertex of the parabola is midway between the focus and the directrix. The distance between the focus and the directrix is $13 - (-1) = 14$, so the vertex is $14 / 2 = 7$ units above the directrix, placing it at $y = -1 + 7 = 6$. The x-coordinate of the vertex is the same as the focus, so the vertex is at (8, 6). The standard form of a parabola that opens upwards or downwards 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), which is 7 in this case. Thus, $4p = 4*7 = 28$, and the equation becomes $(x - 8)^2 = 28(y - 6)$, which matches choice A.

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

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