To find the coordinates of the focus and the equation of the directrix for the parabola given by [tex]\( y^2 = -4x \)[/tex], let's follow these steps:
1. Identify the given equation of the parabola and compare it with the general formula:
The given equation is [tex]\( y^2 = -4x \)[/tex].
The general form of a horizontal parabola opening to the left (or right) is [tex]\( y^2 = 4px \)[/tex].
2. Determine the value of [tex]\( p \)[/tex]:
By comparing [tex]\( y^2 = -4x \)[/tex] to [tex]\( y^2 = 4px \)[/tex], we see that [tex]\( 4p = -4 \)[/tex].
Solving for [tex]\( p \)[/tex]:
[tex]\[
4p = -4 \implies p = -1
\][/tex]
Therefore, the value of [tex]\( p \)[/tex] is [tex]\( -1 \)[/tex].
3. Find the coordinates of the focus:
For the parabola [tex]\( y^2 = 4px \)[/tex], the focus is located at [tex]\( (p, 0) \)[/tex].
Since [tex]\( p = -1 \)[/tex], the coordinates of the focus are:
[tex]\[
(p, 0) = (-1, 0)
\][/tex]
4. Find the equation of the directrix:
The equation of the directrix of the parabola [tex]\( y^2 = 4px \)[/tex] is [tex]\( x = -p \)[/tex].
Since [tex]\( p = -1 \)[/tex], the equation of the directrix is:
[tex]\[
x = -(-1) = 1
\][/tex]
So, the detailed answers are:
- The value of [tex]\( p \)[/tex] is [tex]\( -1 \)[/tex].
- The coordinates of the focus are [tex]\((-1, 0)\)[/tex].
- The equation of the directrix is [tex]\( x = 1 \)[/tex].
