Find the vertex, focus, directrix, and focal width of the parabola.
-$$- \frac { 1 } { 40 } x ^ { 2 } = y$$
A) Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 40
B) Vertex: $( 0,0 )$; Focus: $( 0,10 )$; Directrix: y $= - 10$; Focal width: 10
C) Vertex: $( 0,0 )$; Focus: $( - 20,0 )$; Directrix: $x = 10$; Focal width: 160
D) Vertex: $( 0,0 )$; Focus: $( 0 , - 10 )$; Directrix: $y = 10$; Focal width: 160

Question

Quizplus · Accepted Answer

The given equation is in the form $y = ax^2$, where $a = -\frac{1}{40}$. The vertex of a parabola in this form is at the origin $(0,0)$. The focus is at $(0, \frac{1}{4a})$, which calculates to $(0, -10)$ for $a = -\frac{1}{40}$. The directrix is a line parallel to the x-axis, given by $y = -\frac{1}{4a}$, which is $y = 10$ for $a = -\frac{1}{40}$. The focal width, which is the distance across the parabola through the focus, is $|4a|$, which calculates to 40 for $a = -\frac{1}{40}$.

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