To determine the equation of the directrix for the given parabola [tex]\( y^2 = 12x \)[/tex], we begin by comparing it to the standard form of a parabola that opens horizontally. The standard form of a horizontally opening parabola is [tex]\( y^2 = 4px \)[/tex], where [tex]\( p \)[/tex] is the distance from the vertex to the focus and also from the vertex to the directrix.
Given the parabolic equation:
[tex]\[ y^2 = 12x \][/tex]
1. We compare it to the standard form [tex]\( y^2 = 4px \)[/tex]:
[tex]\[ y^2 = 4px \][/tex]
2. By comparing, we see that [tex]\( 4p = 12 \)[/tex].
3. Solving for [tex]\( p \)[/tex], we divide both sides of the equation by 4:
[tex]\[ p = \frac{12}{4} = 3 \][/tex]
For a horizontally oriented parabola of the form [tex]\( y^2 = 4px \)[/tex]:
- The focus is at [tex]\( (p, 0) \)[/tex]
- The directrix is a vertical line given by [tex]\( x = -p \)[/tex].
With [tex]\( p = 3 \)[/tex], the directrix is:
[tex]\[ x = -3 \][/tex]
Thus, the equation that represents the directrix of the given parabola [tex]\( y^2 = 12x \)[/tex] is:
[tex]\[ x = -3 \][/tex]
So, the correct choice from the given options is:
[tex]\[ x = -3 \][/tex]
