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
Start by rearranging the equation to isolate the \( y^2 \) term on one side. We can rewrite the equation as:
\[ y^2 - 4y = x + 3 \]

### Step 2: Completing the Square
Next, we will complete the square for the \( y \) terms on the left side. To do this, we take the coefficient of \( y \) (which is -4), halve it to get -2, and then square it to get 4. We add and subtract 4 on the left side:
\[ y^2 - 4y + 4 - 4 = x + 3 \]
This simplifies to:
\[ (y - 2)^2 - 4 = x + 3 \]
Now, adding 4 to both sides 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 corresponds to the standard form \( (y - k)^2 = 4a(x - h) \), where \( (h, k) \) is the vertex of the parabola.

### Step 4: Finding the Vertex
From our equation, we can identify:
- \( h = -7 \)
- \( k = 2 \)

Thus, the vertex of the parabola is at the point \( (-7, 2) \).

### Step 5: Finding the Value of \( a \)
In our equation, we see that \( 4a = 1 \), which gives us:
\[ a = \frac{1}{4} \]

### Step 6: Finding the Focus
The focus of a parabola in the form \( (y - k)^2 = 4a(x - h) \) is located at the point \( (h + a, k) \). Therefore, we calculate:
\[ h + a = -7 + \frac{1}{4} = -7 + 0.25 = -\frac{28}{4} + \frac{1}{4} = -\frac{27}{4} \]
Thus, the coordinates of the focus are:
\[ \left(-\frac{27}{4}, 2\right) \]

### Final Answer
The focus of the parabola \( y^2 = x + 4y + 3 \) is:
\[ \left(-\frac{27}{4}, 2\right) \]

---