To find the equation of the directrix for the parabola represented by the equation [tex]\( y^2 = 12x \)[/tex], we can follow these steps:
1. Identify the Standard Form:
The given equation is [tex]\( y^2 = 12x \)[/tex]. This represents a standard form of a parabola that opens to the right. The standard form of a parabola opening rightward is [tex]\( y^2 = 4ax \)[/tex].
2. Determine the Value of [tex]\( a \)[/tex]:
In the standard form [tex]\( y^2 = 4ax \)[/tex], the coefficient of [tex]\( x \)[/tex] (which is 12 in the given equation) is equal to [tex]\( 4a \)[/tex]. Therefore,
[tex]\[
4a = 12
\][/tex]
Solving for [tex]\( a \)[/tex],
[tex]\[
a = \frac{12}{4} = 3
\][/tex]
3. Find the Directrix:
The directrix of a parabola in the form [tex]\( y^2 = 4ax \)[/tex] is given by the equation [tex]\( x = -a \)[/tex]. With [tex]\( a = 3 \)[/tex], the directrix is:
[tex]\[
x = -3
\][/tex]
Hence, the equation that represents the directrix is [tex]\( x = -3 \)[/tex].
So the correct answer is:
[tex]\[
x = -3
\][/tex]
