If the angle `theta` is acute, then the acute angle between `x^(2)(cos theta-sin theta)+2xy cos theta+y^(2)(cos theta+sin theta)=0,` is

Options

A. `2 theta`
B. `theta//3`
C. `theta`
D. `theta//2`

Explanation

To find the acute angle between the given pair of straight lines represented by the equation

\[ x^2 (\cos \theta - \sin \theta) + 2xy \cos \theta + y^2 (\cos \theta + \sin \theta) = 0, \]

we will follow these steps:

### Step 1: Identify coefficients
The general form of a homogeneous equation of the second degree is given by:

\[ ax^2 + 2hxy + by^2 = 0. \]

From the given equation, we can identify the coefficients as:
- \( a = \cos \theta - \sin \theta \)
- \( h = \cos \theta \)
- \( b = \cos \theta + \sin \theta \)

### Step 2: Use the formula for the acute angle between the lines
The acute angle \( \theta' \) between the two lines represented by the equation can be calculated using the formula:

\[ \tan \theta' = \frac{2\sqrt{h^2 - ab}}{a + b}. \]

### Step 3: Calculate \( h^2 - ab \)
First, we need to calculate \( h^2 \) and \( ab \):

\[ h^2 = (\cos \theta)^2 = \cos^2 \theta, \]

\[ ab = (\cos \theta - \sin \theta)(\cos \theta + \sin \theta) = \cos^2 \theta - \sin^2 \theta. \]

Now substituting these into the expression for \( h^2 - ab \):

\[ h^2 - ab = \cos^2 \theta - (\cos^2 \theta - \sin^2 \theta) = \sin^2 \theta. \]

### Step 4: Calculate \( a + b \)
Now we calculate \( a + b \):

\[ a + b = (\cos \theta - \sin \theta) + (\cos \theta + \sin \theta) = 2\cos \theta. \]

### Step 5: Substitute into the formula for \( \tan \theta' \)
Now substituting \( h^2 - ab \) and \( a + b \) into the formula for \( \tan \theta' \):

\[ \tan \theta' = \frac{2\sqrt{\sin^2 \theta}}{2\cos \theta} = \frac{2|\sin \theta|}{2\cos \theta} = \frac{|\sin \theta|}{\cos \theta} = \tan \theta. \]

Since \( \theta \) is acute, \( |\sin \theta| = \sin \theta \).

### Step 6: Conclusion
Thus, we have:

\[ \tan \theta' = \tan \theta. \]

This implies that:

\[ \theta' = \theta. \]

Therefore, the acute angle between the lines is \( \theta \).