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Find the focus and directrix of the following parabola:

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

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

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


Sagot :

To find the focus and directrix of the given parabola [tex]\((y + 3)^2 = 8(x + 3)\)[/tex], we will first rewrite this equation in the standard form of a parabola.

### Step 1: Identify the Standard Form
The standard form of a horizontal parabola that opens to the right is [tex]\((y - k)^2 = 4p(x - h)\)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\(p\)[/tex] is the distance from the vertex to the focus (also to the directrix).

### Step 2: Rewrite the Given Equation
The given equation is:

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

We can rewrite this in the standard form by recognizing that:
[tex]\((y + 3)\)[/tex] is analogous to [tex]\((y - k)\)[/tex], and [tex]\((x + 3)\)[/tex] is analogous to [tex]\((x - h)\)[/tex].

Thus, comparing with [tex]\((y - k)^2 = 4p(x - h)\)[/tex]:

- [tex]\(k = -3\)[/tex]
- [tex]\(h = -3\)[/tex]
- [tex]\(4p = 8\)[/tex]

### Step 3: Solve for [tex]\(p\)[/tex]
From the equation [tex]\(4p = 8\)[/tex], we can solve for [tex]\(p\)[/tex]:

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

### Step 4: Find the Focus
The focus of a parabola in this form is located at [tex]\((h + p, k)\)[/tex].

Substituting the known values of [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(p\)[/tex]:

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

Thus, the coordinates of the focus are [tex]\((-1, -3)\)[/tex].

### Step 5: Find the Directrix
The directrix of the parabola in this form is given by the vertical line:

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

Substituting the known values of [tex]\(h\)[/tex] and [tex]\(p\)[/tex]:

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

Thus, the equation of the directrix is [tex]\(x = -5\)[/tex].

### Final Answer

- Focus: [tex]\((-1, -3)\)[/tex]
- Directrix: [tex]\(x = -5\)[/tex]

So, the answer to the question is:

Focus: ([–1], [tex]\(-3\)[/tex])
Directrix: [tex]\(x = -5\)[/tex]