To solve for the directrix and the focus of the parabola given by the equation [tex]\( y^2 = -24x \)[/tex], let's follow the steps:
### Step-by-Step Solution:
1. Identify the Standard Form:
The given equation of the parabola is [tex]\( y^2 = -24x \)[/tex]. This can be compared with the standard form of the parabolas of the type [tex]\( y^2 = 4ax \)[/tex].
2. Rewrite to Identify 'a':
[tex]\( y^2 = -24x \)[/tex] can be written as [tex]\( y^2 = 4(-6)x \)[/tex].
Here, we compare it with [tex]\( y^2 = 4ax \)[/tex] and identify that [tex]\( 4a = -24 \)[/tex], therefore [tex]\( a = -6 \)[/tex].
3. Determine the Focus:
For a parabola of the form [tex]\( y^2 = 4ax \)[/tex], the focus is at the point [tex]\( (a, 0) \)[/tex].
Substituting [tex]\( a = -6 \)[/tex], the focus is at [tex]\( (-6, 0) \)[/tex].
4. Determine the Directrix:
The directrix of a parabola of the form [tex]\( y^2 = 4ax \)[/tex] is given by the line [tex]\( x = -a \)[/tex].
Substituting [tex]\( a = -6 \)[/tex], the directrix is given by [tex]\( x = -(-6) \)[/tex], which simplifies to [tex]\( x = 6 \)[/tex].
### Conclusion:
- Focus of the parabola: [tex]\( (-6, 0) \)[/tex]
- Equation of the directrix: [tex]\( x = 6 \)[/tex]
Thus, 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].
