To determine the equation that represents the directrix of the given parabola, we need to start by analyzing the standard form of the parabola.
The given equation of the parabola is:
[tex]\[ y^2 = 12x \][/tex]
1. Identify the standard form:
The standard form for a parabola that opens to the right is:
[tex]\[ y^2 = 4px \][/tex]
where [tex]\( p \)[/tex] is the distance from the vertex to the focus (or from the vertex to the directrix, but in the opposite direction).
2. Compare to find [tex]\( p \)[/tex]:
By comparing the given equation [tex]\( y^2 = 12x \)[/tex] with the standard form [tex]\( y^2 = 4px \)[/tex], we can identify the value of [tex]\( p \)[/tex]:
[tex]\[ 4p = 12 \][/tex]
Solving for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{12}{4} \][/tex]
[tex]\[ p = 3 \][/tex]
3. Determine the directrix:
The directrix of a parabola given in the form [tex]\( y^2 = 4px \)[/tex] is a vertical line located at [tex]\( x = -p \)[/tex].
Since [tex]\( p = 3 \)[/tex]:
[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]\[ x = -3 \][/tex]
