To find the equation of the directrix for the parabola given by the equation [tex]\( y^2 = -24x \)[/tex], we can follow these steps:
1. Identify the standard form: The given equation [tex]\( y^2 = -24x \)[/tex] is in the form [tex]\( y^2 = 4px \)[/tex]. The standard form for a horizontally oriented parabola is [tex]\( y^2 = 4px \)[/tex], where [tex]\( p \)[/tex] is the distance from the vertex to the focus, and also to the directrix but in the opposite direction.
2. Determine the value of [tex]\( p \)[/tex]: By comparing [tex]\( y^2 = -24x \)[/tex] to the standard form [tex]\( y^2 = 4px \)[/tex]:
[tex]\[
4p = -24
\][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{-24}{4} = -6
\][/tex]
3. Find the directrix: For a parabola in the form [tex]\( y^2 = 4px \)[/tex], the directrix is given by [tex]\( x = -p \)[/tex]. Substituting the value of [tex]\( p \)[/tex]:
[tex]\[
x = -(-6) = 6
\][/tex]
The equation of the directrix is [tex]\( x = 6 \)[/tex].
Hence, the correct answer is:
D. [tex]\( x=6 \)[/tex]
