The focus of the parabola `y^2=x+4y+3`, is :

Options

A. `(-7,(13)/(4))`
B. `(-7,(9)/(4))`
C. `((-27)/(4),2)`
D. `((-15)/(2),2)`

Answer

To find the focus of the parabola given by the equation \( y^2 = x + 4y + 3 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ y^2 = x + 4y + 3 \]
We can rearrange this to isolate \( x \):
\[ y^2 - 4y = x + 3 \]
Subtracting 3 from both sides gives us:
\[ y^2 - 4y - 3 = x \]

### Step 2: Completing the Square
Next, we need to complete the square for the \( y \) terms on the left side. We take the coefficient of \( y \) (which is -4), halve it to get -2, and square it to get 4. We add and subtract 4:
\[ y^2 - 4y + 4 - 4 - 3 = x \]
This simplifies to:
\[ (y - 2)^2 - 7 = x \]
Rearranging gives:
\[ (y - 2)^2 = x + 7 \]

### Step 3: Identifying the Standard Form
Now, we can rewrite the equation in the standard form of a parabola:
\[ (y - 2)^2 = 1(x + 7) \]
This matches the standard form \( (y - k)^2 = 4a(x - h) \), where \( (h, k) \) is the vertex of the parabola.

### Step 4: Finding the Vertex
From the equation \( (y - 2)^2 = 1(x + 7) \), we can identify:
- Vertex \( (h, k) = (-7, 2) \)
- Here, \( 4a = 1 \), so \( a = \frac{1}{4} \).

### Step 5: Finding the Focus
For a parabola that opens to the right, the focus is located at:
\[ (h + a, k) \]
Substituting the values we found:
\[ h = -7, \quad k = 2, \quad a = \frac{1}{4} \]
Thus, the coordinates of the focus are:
\[ \left(-7 + \frac{1}{4}, 2\right) = \left(-\frac{28}{4} + \frac{1}{4}, 2\right) = \left(-\frac{27}{4}, 2\right) \]

### Final Answer
The focus of the parabola is:
\[ \left(-\frac{27}{4}, 2\right) \]

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