The acute angle `theta` between the lines represented by `3x^(2)+2xy-y^(2)=0` is

Options

A. `tan^(-1)((1)/(2))`
B. `tan^(-1)((3)/(2))`
C. `tan^(-1)((2)/(3))`
D. `tan^(-1)2`

Answer

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

### Step 1: Identify coefficients
The general form of the equation of a pair of straight lines is given by:
\[ Ax^2 + 2Hxy + By^2 = 0 \]
From the given equation \( 3x^2 + 2xy - y^2 = 0 \), we can identify:
- \( A = 3 \)
- \( B = -1 \)
- \( H = 1 \)

### Step 2: Use the formula for \( \tan \theta \)
The formula for the tangent of the angle \( \theta \) between the two lines is:
\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]
Substituting the values of \( A \), \( B \), and \( H \):
\[ \tan \theta = \frac{2\sqrt{1^2 - (3)(-1)}}{3 + (-1)} \]

### Step 3: Calculate \( H^2 - AB \)
Calculate \( H^2 - AB \):
\[ H^2 - AB = 1 - (3)(-1) = 1 + 3 = 4 \]

### Step 4: Substitute back into the formula
Now substitute this back into the formula for \( \tan \theta \):
\[ \tan \theta = \frac{2\sqrt{4}}{3 - 1} = \frac{2 \times 2}{2} = \frac{4}{2} = 2 \]

### Step 5: Find \( \theta \)
To find \( \theta \), we take the inverse tangent:
\[ \theta = \tan^{-1}(2) \]

### Final Answer
Thus, the acute angle \( \theta \) between the lines is:
\[ \theta = \tan^{-1}(2) \]

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