To find the coordinates of the focus and the equation of the directrix for the parabola given by [tex]\( y^2 = -4x \)[/tex], we will follow these steps.
### Step-by-Step Solution
1. Identify the form of the given equation:
The given equation of the parabola is [tex]\( y^2 = -4x \)[/tex].
2. Compare it with the standard form:
The standard form of a parabola that opens leftward is [tex]\( y^2 = 4px \)[/tex].
3. Determine the value of [tex]\( p \)[/tex]:
By comparing the given equation [tex]\( y^2 = -4x \)[/tex] with [tex]\( y^2 = 4px \)[/tex]:
- We can see that [tex]\( 4p = -4 \)[/tex].
- This implies [tex]\( p = \frac{-4}{4} = -1 \)[/tex].
4. Find the coordinates of the focus:
For the standard form [tex]\( y^2 = 4px \)[/tex], the coordinates of the focus are [tex]\( (p, 0) \)[/tex].
- Since [tex]\( p = -1 \)[/tex], the coordinates of the focus are [tex]\( (-1, 0) \)[/tex].
5. Find the equation of the directrix:
The equation of the directrix for the standard form [tex]\( y^2 = 4px \)[/tex] is [tex]\( x = -p \)[/tex].
- Since [tex]\( p = -1 \)[/tex], the equation of the directrix is [tex]\( x = -(-1) = x = 1 \)[/tex].
### Final Results
- 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].
Using these values, we have:
- The general formula for this parabola is [tex]\( y^2 = 4px \)[/tex].
Therefore, the value of [tex]\( p \)[/tex] is [tex]\(\boxed{-1}\)[/tex].
- The coordinates of the focus are [tex]\(\boxed{(-1, 0)}\)[/tex].
- The equation of the directrix is [tex]\(\boxed{x = 1}\)[/tex].
