the equation of the axes of the ellispe `3x^(2)+4y^(2)+6x-8y-5=0` are

Options

A. `x-3,y=5`
B. `x+3=0,y-5=0`
C. `x-1=0,y=0`
D. `x+1=0,y-1=0`

Answer

To find the equations of the axes of the ellipse given by the equation \(3x^2 + 4y^2 + 6x - 8y - 5 = 0\), we will follow these steps:

### Step 1: Rearranging the equation
We start by rearranging the given equation:
\[ 3x^2 + 4y^2 + 6x - 8y - 5 = 0 \]
We can rewrite it as:
\[ 3x^2 + 6x + 4y^2 - 8y = 5 \]

### Step 2: Completing the square for \(x\)
Next, we complete the square for the \(x\) terms:
\[ 3(x^2 + 2x) + 4y^2 - 8y = 5 \]
To complete the square for \(x^2 + 2x\), we take half of the coefficient of \(x\) (which is 2), square it (resulting in 1), and add and subtract it inside the parentheses:
\[ 3((x^2 + 2x + 1) - 1) + 4y^2 - 8y = 5 \]
This simplifies to:
\[ 3((x + 1)^2 - 1) + 4y^2 - 8y = 5 \]
Expanding this gives:
\[ 3(x + 1)^2 - 3 + 4y^2 - 8y = 5 \]
Rearranging yields:
\[ 3(x + 1)^2 + 4y^2 - 8y = 8 \]

### Step 3: Completing the square for \(y\)
Now we complete the square for the \(y\) terms:
\[ 4(y^2 - 2y) = 4((y^2 - 2y + 1) - 1) = 4((y - 1)^2 - 1) \]
Substituting this back into the equation gives:
\[ 3(x + 1)^2 + 4((y - 1)^2 - 1) = 8 \]
This simplifies to:
\[ 3(x + 1)^2 + 4(y - 1)^2 - 4 = 8 \]
Rearranging yields:
\[ 3(x + 1)^2 + 4(y - 1)^2 = 12 \]

### Step 4: Dividing by 12
To express this in standard form, we divide the entire equation by 12:
\[ \frac{3(x + 1)^2}{12} + \frac{4(y - 1)^2}{12} = 1 \]
This simplifies to:
\[ \frac{(x + 1)^2}{4} + \frac{(y - 1)^2}{3} = 1 \]

### Step 5: Identifying the axes
From the standard form of the ellipse \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\), we identify:
- Center \((h, k) = (-1, 1)\)
- \(a^2 = 4\) (so \(a = 2\))
- \(b^2 = 3\) (so \(b = \sqrt{3}\))

Since \(a^2 > b^2\), the major axis is horizontal, and the minor axis is vertical.

### Step 6: Writing the equations of the axes
The equations of the axes are:
- Major axis (horizontal): \(y = 1\)
- Minor axis (vertical): \(x = -1\)

Thus, the equations of the axes of the ellipse are:
\[ \text{Major Axis: } y = 1 \]
\[ \text{Minor Axis: } x = -1 \]