The focus of the conic `x ^(2) -6x + 4y +1=0` is

Options

A. `(2,3)`
B. `(3,2)`
C. `(3,1)`
D. `(1,4)`

Answer

To find the focus of the conic given by the equation \(x^2 - 6x + 4y + 1 = 0\), we will convert it into the standard form of a parabola and then determine the coordinates of the focus.

### Step-by-Step Solution:

1. **Rearranging the Equation:**
Start with the given equation:
\[ x^2 - 6x + 4y + 1 = 0 \]
Rearranging it gives:
\[ x^2 - 6x + 4y = -1 \]

2. **Completing the Square:**
Focus on the \(x\) terms \(x^2 - 6x\). To complete the square:
- Take half of the coefficient of \(x\) (which is \(-6\)), square it, and add/subtract it:
\[ \left(-\frac{6}{2}\right)^2 = 9 \]
Add and subtract \(9\):
\[ x^2 - 6x + 9 - 9 + 4y = -1 \]
This simplifies to:
\[ (x - 3)^2 - 9 + 4y = -1 \]
Rearranging gives:
\[ (x - 3)^2 + 4y - 9 = -1 \]
Thus:
\[ (x - 3)^2 + 4y = 8 \]

3. **Isolating \(y\):**
Now isolate \(y\):
\[ 4y = 8 - (x - 3)^2 \]
Divide everything by \(4\):
\[ y = 2 - \frac{1}{4}(x - 3)^2 \]

4. **Standard Form of the Parabola:**
The equation can now be expressed in the standard form of a parabola:
\[ (x - h)^2 = 4p(y - k) \]
Here, we have:
\[ (x - 3)^2 = -4(y - 2) \]
From this, we can identify:
- \(h = 3\)
- \(k = 2\)
- \(4p = -4\) which gives \(p = -1\)

5. **Finding the Focus:**
The coordinates of the focus are given by \((h, k + p)\):
\[ \text{Focus} = (3, 2 + (-1)) = (3, 1) \]

### Final Answer:
The focus of the conic \(x^2 - 6x + 4y + 1 = 0\) is \((3, 1)\).