The length of the axes of the conic `9x^2 + 4y^2 - 6x + 4y +1=0`, are

Options

A. `(1)/(2),9`
B. `3,(2)/(5)`
C. `1,(2)/(3)`
D. `3,2`

Explanation

To find the lengths of the axes of the conic given by the equation \(9x^2 + 4y^2 - 6x + 4y + 1 = 0\), we will follow these steps:

### Step 1: Rearranging the equation
We start with the given equation:
\[ 9x^2 + 4y^2 - 6x + 4y + 1 = 0 \]
We can rearrange it to:
\[ 9x^2 - 6x + 4y^2 + 4y + 1 = 0 \]
Next, we will move the constant term to the other side:
\[ 9x^2 - 6x + 4y^2 + 4y = -1 \]

### Step 2: Completing the square for \(x\) and \(y\)
Now, we will complete the square for the \(x\) and \(y\) terms.

**For \(x\):**
The terms involving \(x\) are \(9x^2 - 6x\). We factor out 9:
\[ 9(x^2 - \frac{2}{3}x) \]
To complete the square, we take half of \(-\frac{2}{3}\) (which is \(-\frac{1}{3}\)), square it (getting \(\frac{1}{9}\)), and add and subtract it inside the parentheses:
\[ 9\left(x^2 - \frac{2}{3}x + \frac{1}{9} - \frac{1}{9}\right) = 9\left((x - \frac{1}{3})^2 - \frac{1}{9}\right) \]
This simplifies to:
\[ 9(x - \frac{1}{3})^2 - 1 \]

**For \(y\):**
The terms involving \(y\) are \(4y^2 + 4y\). We factor out 4:
\[ 4(y^2 + y) \]
To complete the square, we take half of \(1\) (which is \(\frac{1}{2}\)), square it (getting \(\frac{1}{4}\)), and add and subtract it inside the parentheses:
\[ 4\left(y^2 + y + \frac{1}{4} - \frac{1}{4}\right) = 4\left((y + \frac{1}{2})^2 - \frac{1}{4}\right) \]
This simplifies to:
\[ 4(y + \frac{1}{2})^2 - 1 \]

### Step 3: Substitute back into the equation
Now substituting back, we have:
\[ 9\left((x - \frac{1}{3})^2 - \frac{1}{9}\right) + 4\left((y + \frac{1}{2})^2 - \frac{1}{4}\right) = -1 \]
This simplifies to:
\[ 9(x - \frac{1}{3})^2 - 1 + 4(y + \frac{1}{2})^2 - 1 = -1 \]
Combining the constants:
\[ 9(x - \frac{1}{3})^2 + 4(y + \frac{1}{2})^2 - 2 = -1 \]
Adding 2 to both sides:
\[ 9(x - \frac{1}{3})^2 + 4(y + \frac{1}{2})^2 = 1 \]

### Step 4: Dividing by the right-hand side
Now we divide the entire equation by 1:
\[ \frac{9(x - \frac{1}{3})^2}{1} + \frac{4(y + \frac{1}{2})^2}{1} = 1 \]
This can be rewritten as:
\[ \frac{(x - \frac{1}{3})^2}{\frac{1}{9}} + \frac{(y + \frac{1}{2})^2}{\frac{1}{4}} = 1 \]

### Step 5: Identifying parameters \(a\) and \(b\)
From the standard form of the ellipse:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
we can identify:
- \(a^2 = \frac{1}{9} \Rightarrow a = \frac{1}{3}\)
- \(b^2 = \frac{1}{4} \Rightarrow b = \frac{1}{2}\)

### Step 6: Finding lengths of the axes
The lengths of the axes are given by:
- Length of the major axis = \(2b = 2 \times \frac{1}{2} = 1\)
- Length of the minor axis = \(2a = 2 \times \frac{1}{3} = \frac{2}{3}\)

### Final Answer
Thus, the lengths of the axes of the ellipse are:
- Major axis: 1
- Minor axis: \(\frac{2}{3}\)