To determine the equation of the directrix of the parabola represented by [tex]\( y^2 = 12x \)[/tex], let's follow these steps:
1. Identify the form of the parabola:
The standard form of a parabola that opens either to the right or left is [tex]\( y^2 = 4ax \)[/tex], where [tex]\( a \)[/tex] is a constant.
2. Match the given equation to the standard form:
Compare the given equation [tex]\( y^2 = 12x \)[/tex] to [tex]\( y^2 = 4ax \)[/tex] to find the value of [tex]\( a \)[/tex].
- Here, [tex]\( 4a = 12 \)[/tex].
3. Solve for [tex]\( a \)[/tex]:
To find [tex]\( a \)[/tex], divide both sides of the equation [tex]\( 4a = 12 \)[/tex] by 4:
[tex]\[ a = \frac{12}{4} \][/tex]
[tex]\[ a = 3 \][/tex]
4. Determine the directrix:
For a parabola of the form [tex]\( y^2 = 4ax \)[/tex], the directrix is given by [tex]\( x = -a \)[/tex].
5. Substitute the value of [tex]\( a \)[/tex]:
Substitute [tex]\( a = 3 \)[/tex] into the directrix equation:
[tex]\[ x = -3 \][/tex]
Therefore, the equation that represents the directrix of the given parabola [tex]\( y^2 = 12x \)[/tex] is
[tex]\[ x = -3. \][/tex]
So, the correct answer is:
[tex]\[ \boxed{x = -3} \][/tex]
