Find the vertex, focus, directrix, and focal width of the parabola.
-$$( x - 9 ) ^ { 2 } = - 8 ( y + 6 )$$
A) Vertex: $( - 6,9 )$; Focus: $( - 8,9 )$; Directrix: $x = 11$; Focal width: 2
B) Vertex: $( - 9,6 )$; Focus: $( - 9 , - 2 )$; Directrix: $y = 14$; Focal width: 8
C) Vertex: $( - 6,9 )$; Focus: $( - 14,9 )$; Directrix: $x = 17$; Focal width: 8
D) Vertex: $( 9 , - 6 )$; Focus: $( 9 , - 8 )$; Directrix: $y = - 4$; Focal width: 8

Question

Quizplus · Accepted Answer

The given equation is in the form $(x - h)^2 = 4p(y - k)$, where $(h, k)$ is the vertex. Comparing, we find the vertex is $(9, -6)$. The value of $4p = -8$, so $p = -2$, indicating the parabola opens downwards. The focus is $(h, k + p) = (9, -6 - 2) = (9, -8)$, the directrix is $y = k - p = -6 + 2 = -4$, and the focal width, which is the absolute value of $4p$, is $8$.

Find the Vertex, Focus, Directrix, and Focal Width of the Parabola