Answer :
To find the equation of the directrix for the given parabola [tex]\( y^2 = -24x \)[/tex], we first recognize the general form of the given equation.
The standard form of a parabola that opens to the left or right is [tex]\( y^2 = -4ax \)[/tex] (for parabolas that open to the left). Here, comparing [tex]\( y^2 = -24x \)[/tex] with [tex]\( y^2 = -4ax \)[/tex], we identify that:
[tex]\[ -4a = -24 \][/tex]
By solving this equation for [tex]\( a \)[/tex],
[tex]\[ 4a = 24 \][/tex]
[tex]\[ a = \frac{24}{4} \][/tex]
[tex]\[ a = 6 \][/tex]
For a parabola of the form [tex]\( y^2 = -4ax \)[/tex], the directrix is a vertical line given by the equation [tex]\( x = a \)[/tex].
Substituting [tex]\( a \)[/tex]:
The directrix is [tex]\( x = 6 \)[/tex].
Therefore, the correct answer to the question is:
[tex]\[ x = 6 \][/tex]
However, the answer choices provided include [tex]\( y = 4 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( y = 6 \)[/tex], but we see that there is no exact match. This indicates a possible error in the provided choices or question. Based on standard mathematical conventions, the correct form derived should logically be [tex]\( x = 6 \)[/tex]. If we must choose from the given options, then none of them are correct as per standard mathematical derivations.
The standard form of a parabola that opens to the left or right is [tex]\( y^2 = -4ax \)[/tex] (for parabolas that open to the left). Here, comparing [tex]\( y^2 = -24x \)[/tex] with [tex]\( y^2 = -4ax \)[/tex], we identify that:
[tex]\[ -4a = -24 \][/tex]
By solving this equation for [tex]\( a \)[/tex],
[tex]\[ 4a = 24 \][/tex]
[tex]\[ a = \frac{24}{4} \][/tex]
[tex]\[ a = 6 \][/tex]
For a parabola of the form [tex]\( y^2 = -4ax \)[/tex], the directrix is a vertical line given by the equation [tex]\( x = a \)[/tex].
Substituting [tex]\( a \)[/tex]:
The directrix is [tex]\( x = 6 \)[/tex].
Therefore, the correct answer to the question is:
[tex]\[ x = 6 \][/tex]
However, the answer choices provided include [tex]\( y = 4 \)[/tex], [tex]\( x = 4 \)[/tex], and [tex]\( y = 6 \)[/tex], but we see that there is no exact match. This indicates a possible error in the provided choices or question. Based on standard mathematical conventions, the correct form derived should logically be [tex]\( x = 6 \)[/tex]. If we must choose from the given options, then none of them are correct as per standard mathematical derivations.
