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Find the focus and directrix of the following parabola:

[tex]\[ (y + 3)^2 = 8(x + 3) \][/tex]

Focus: [tex]\((\square, \square)\)[/tex]

Directrix: [tex]\(x = \square\)[/tex]


Sagot :

To find the focus and directrix of the parabola given by the equation:

[tex]\[ (y + 3)^2 = 8(x + 3) \][/tex]

we will follow these steps:

### Step 1: Rewrite the Given Parabola Equation

First, let's rewrite the given equation in a more familiar form. The given equation is:

[tex]\[ (y + 3)^2 = 8(x + 3) \][/tex]

### Step 2: Identify the Standard Form and Parameters

We recognize that this equation resembles the standard form of a parabola that opens horizontally:

[tex]\[ (y - k)^2 = 4p(x - h) \][/tex]

### Step 3: Compare with Standard Form

By comparing:

[tex]\[ (y + 3)^2 = 8(x + 3) \][/tex]

to:

[tex]\[ (y - k)^2 = 4p(x - h) \][/tex]

we can see that:
- [tex]\( k = -3 \)[/tex]
- [tex]\( h = -3 \)[/tex]
- [tex]\( 4p = 8 \)[/tex]

### Step 4: Solve for 'p'

Now, solve for [tex]\( p \)[/tex]:

[tex]\[ 4p = 8 \implies p = \frac{8}{4} = 2 \][/tex]

### Step 5: Determine the Coordinates of the Focus

The focus of the parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex] is given by the coordinates [tex]\((h + p, k)\)[/tex].

With the values substituted in:

[tex]\[ h = -3, \quad k = -3, \quad p = 2 \][/tex]

So the coordinates of the focus are:

[tex]\[ (h + p, k) = (-3 + 2, -3) = (-1, -3) \][/tex]

### Step 6: Find the Equation of the Directrix

The directrix of the parabola [tex]\((y - k)^2 = 4p(x - h)\)[/tex] has the equation:

[tex]\[ x = h - p \][/tex]

With the values substituted in:

[tex]\[ h = -3, \quad p = 2 \][/tex]

The equation of the directrix is:

[tex]\[ x = -3 - 2 = -5 \][/tex]

### Final Answers

The focus of the given parabola is:

[tex]\[ (-1, -3) \][/tex]

The directrix of the given parabola is:

[tex]\[ x = -5 \][/tex]

So, the final answers are:

Focus: [tex]\((-1, -3)\)[/tex]

Directrix: [tex]\(x = -5\)[/tex]