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Find the coordinates of the focus and the equation of the directrix for the parabola given by [tex]y^2 = -4x[/tex].

The general formula for this parabola is [tex]y^2 = 4px[/tex].

Therefore, the value of [tex]p[/tex] is [tex]\(\square\)[/tex].

The coordinates of the focus are [tex]\(\qquad\)[/tex].

The equation of the directrix is [tex]\(\qquad\)[/tex].


Sagot :

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].