If the equation of the pair of straight lines passing through the point `(1,1)` , one making an angle `theta` with the positive direction of the x-axis and the other making the same angle with the positive direction of the y-axis, is `x^2-(a+2)x y+y^2+a(x+y-1)=0,a!=2,` then the value of `sin2theta` is

Options

A. `a-2`
B. `a+2`
C. `2//(a+2)`
D. `2//a`

Explanation

To solve the problem step by step, we will analyze the given equation of the pair of straight lines and derive the value of \( \sin 2\theta \).

### Step 1: Understand the given equation
The equation of the pair of straight lines is given as:
\[ x^2 - (a + 2)xy + y^2 + a(x + y - 1) = 0 \]
This represents a pair of straight lines passing through the point \((1, 1)\).

### Step 2: Rewrite the equation
We can rewrite the equation to make it easier to analyze:
\[ x^2 - (a + 2)xy + y^2 + ax + ay - a = 0 \]

### Step 3: Identify the slopes of the lines
The lines make angles \( \theta \) with the positive x-axis and y-axis. The slopes of these lines can be represented as:
- \( m_1 = \tan(\theta) \) for the line making angle \( \theta \) with the x-axis.
- \( m_2 = \cot(\theta) \) for the line making angle \( \theta \) with the y-axis.

### Step 4: Form the joint equation of the lines
The joint equation of the lines can be expressed as:
\[ (y - 1) - \tan(\theta)(x - 1) = 0 \]
\[ (y - 1) - \cot(\theta)(x - 1) = 0 \]

### Step 5: Expand the joint equation
Expanding the joint equation gives us:
\[ (y - 1)^2 - (\cot(\theta) + \tan(\theta))(y - 1)(x - 1) + (x - 1)^2 = 0 \]

### Step 6: Compare coefficients
Now, we can compare coefficients of the expanded equation with the original equation. We have:
- Coefficient of \(xy\): \(-(\cot(\theta) + \tan(\theta))\)
- Coefficient of \(x + y\): \( \cot(\theta) + \tan(\theta) - 2 \)

### Step 7: Relate coefficients to \(a\)
From the original equation, we know:
\[ -(a + 2) = -(\cot(\theta) + \tan(\theta)) \]
This gives us:
\[ \cot(\theta) + \tan(\theta) = a + 2 \]

### Step 8: Use the identity for \(\sin 2\theta\)
We know that:
\[ \cot(\theta) + \tan(\theta) = \frac{1}{\sin(\theta)\cos(\theta)} \]
Thus:
\[ \cot(\theta) + \tan(\theta) = \frac{2}{\sin(2\theta)} \]

### Step 9: Set up the equation
Equating the two expressions we derived:
\[ \frac{2}{\sin(2\theta)} = a + 2 \]

### Step 10: Solve for \(\sin 2\theta\)
Rearranging gives:
\[ \sin(2\theta) = \frac{2}{a + 2} \]

### Final Answer
Thus, the value of \( \sin 2\theta \) is:
\[ \sin 2\theta = \frac{2}{a + 2} \]

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