To remove xy term from the second degree equation `5x^2 + 8xy + 5y^2 + 3x + 2y + 5 = 0` , the coordinates axes are rotated through an angle `theta` , then` theta` equals -

Options

A. `pi//2`
B. `pi//4`
C. `3pi//4`
D. `pi//8`

Answer

To remove the \(xy\) term from the second-degree equation \(5x^2 + 8xy + 5y^2 + 3x + 2y + 5 = 0\), we need to determine the angle \(\theta\) through which the coordinate axes should be rotated. The formula to find \(\theta\) is given by:

\[ \tan(2\theta) = \frac{2h}{a - b} \]

where:
- \(h\) is the coefficient of \(xy\),
- \(a\) is the coefficient of \(x^2\),
- \(b\) is the coefficient of \(y^2\).

### Step 1: Identify the coefficients
From the given equation \(5x^2 + 8xy + 5y^2 + 3x + 2y + 5 = 0\), we can identify:
- \(a = 5\) (coefficient of \(x^2\)),
- \(b = 5\) (coefficient of \(y^2\)),
- \(h = 4\) (coefficient of \(xy\), which is half of 8).

### Step 2: Substitute the values into the formula
Now, we substitute these values into the formula:

\[ \tan(2\theta) = \frac{2h}{a - b} = \frac{2 \cdot 4}{5 - 5} \]

### Step 3: Simplify the expression
Calculating the denominator:

\[ 5 - 5 = 0 \]

This results in:

\[ \tan(2\theta) = \frac{8}{0} \]

### Step 4: Analyze the result
Since division by zero is undefined, this implies that \(2\theta\) is equal to \(\frac{\pi}{2}\) (or \(90^\circ\)), leading to:

\[ \theta = \frac{\pi}{4} \text{ (or } 45^\circ\text{)} \]

### Conclusion
Thus, the angle \(\theta\) through which the coordinate axes should be rotated to eliminate the \(xy\) term is:

\[ \theta = \frac{\pi}{4} \text{ (or } 45^\circ\text{)} \]