To find the equation of the directrix of a parabola that has its vertex at the origin [tex]\((0,0)\)[/tex] and its focus at [tex]\((-2,0)\)[/tex], we need to understand the general properties of parabolas.
1. The vertex of the parabola is at the origin [tex]\((0,0)\)[/tex].
2. The focus of the parabola is located at [tex]\((-2,0)\)[/tex].
For a parabola with its vertex at the origin, the equation of the parabola is determined by the placement of the focus:
- If the focus is to the left or the right of the vertex (along the x-axis), the parabola opens horizontally.
- If the focus is above or below the vertex (along the y-axis), the parabola opens vertically.
In this case, since the focus [tex]\((-2,0)\)[/tex] is to the left of the vertex, the parabola opens horizontally.
For a horizontal parabola with vertex [tex]\((0,0)\)[/tex] and focus at [tex]\((a,0)\)[/tex]:
- The equation of the directrix is [tex]\(x = -a\)[/tex], where [tex]\(a\)[/tex] is the distance from the vertex to the focus.
Here, the focus is [tex]\((-2,0)\)[/tex], so [tex]\(a = -2\)[/tex]. Therefore, the directrix is [tex]\(x = 2\)[/tex].
So, the equation for the directrix is:
[tex]$x = 2$[/tex]
