The angle between the pair of lines `2x^(2)+5xy+2y^(2)-3x-3y+1=0` is

Options

A. `tan^(-1)((4)/(5))`
B. `sin^(-1)((4)/(5))`
C. `cos^(-1)((4)/(5))`
D. `tan^(-1)((3)/(4))`

Answer

To find the angle between the pair of lines represented by the equation \(2x^2 + 5xy + 2y^2 - 3x - 3y + 1 = 0\), we can follow these steps:

### Step 1: Identify the coefficients from the given equation
The general form of the equation for a pair of straight lines is:
\[ ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0 \]
From the given equation \(2x^2 + 5xy + 2y^2 - 3x - 3y + 1 = 0\), we can identify:
- \(a = 2\)
- \(b = 2\)
- \(h = \frac{5}{2}\) (since \(2h = 5\))
- \(g = -\frac{3}{2}\) (since \(2g = -3\))
- \(f = -\frac{3}{2}\) (since \(2f = -3\))
- \(c = 1\)

### Step 2: Use the formula for the tangent of the angle between the lines
The formula for the tangent of the angle \(\theta\) between the pair of lines is given by:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]

### Step 3: Calculate \(h^2 - ab\)
First, we need to calculate \(h^2\) and \(ab\):
- \(h^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}\)
- \(ab = 2 \cdot 2 = 4\)

Now, we can find \(h^2 - ab\):
\[ h^2 - ab = \frac{25}{4} - 4 = \frac{25}{4} - \frac{16}{4} = \frac{9}{4} \]

### Step 4: Substitute values into the tangent formula
Now, substitute the values into the formula:
\[ \tan \theta = \frac{2\sqrt{\frac{9}{4}}}{2 + 2} = \frac{2 \cdot \frac{3}{2}}{4} = \frac{3}{4} \]

### Step 5: Find the angle \(\theta\)
To find the angle \(\theta\), we take the inverse tangent:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

### Final Answer
Thus, the angle between the pair of lines is:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

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