To solve the problem, let's break it down step-by-step:
1. Identify the form of the given equation:
The given equation of the parabola is [tex]\( y^2 = -24x \)[/tex].
2. Rewrite the equation in the standard form:
The standard form of a parabola that opens left or right is [tex]\( y^2 = 4px \)[/tex]. By comparing this with the given equation [tex]\( y^2 = -24x \)[/tex], we can identify that [tex]\( 4p = -24 \)[/tex].
3. Solve for [tex]\( p \)[/tex]:
Dividing both sides by 4, we get:
[tex]\[
p = \frac{-24}{4} = -6
\][/tex]
4. Determine the equation of the directrix:
For a parabola in the form [tex]\( y^2 = 4px \)[/tex], the directrix is given by [tex]\( x = -p \)[/tex]. Substituting [tex]\( p = -6 \)[/tex], we get:
[tex]\[
\text{Directrix: } x = -(-6) = 6
\][/tex]
5. Determine the focus:
The focus of a parabola given by [tex]\( y^2 = 4px \)[/tex] is located at [tex]\( (p, 0) \)[/tex]. Substituting [tex]\( p = -6 \)[/tex], we get:
[tex]\[
\text{Focus: } (-6, 0)
\][/tex]
So, the correct answers are:
- The equation of the directrix of the parabola is [tex]\( x = 6 \)[/tex].
- The focus of the parabola is [tex]\( (-6, 0) \)[/tex].
