The set of all values of `a^(2)` for which the line `x+y=0` bisects two distinct chords drawn from a point `P((1+a)/(2),(1-a)/(2))` on the circle `2x^(2)+2y^(2)-(1+a)x-(1-a)y=0`,is equal to :

Options

A. `(8,oo)`
B. `(0,4]`
C. `(4,oo)`
D. `(2,12]`

Answer

To solve the problem, we need to determine the set of all values of \( a^2 \) for which the line \( x + y = 0 \) bisects two distinct chords drawn from the point \( P\left(\frac{1+a}{2}, \frac{1-a}{2}\right) \) on the circle defined by the equation \( 2x^2 + 2y^2 - (1+a)x - (1-a)y = 0 \).

### Step-by-step Solution:

1. **Rewrite the Circle Equation**:
The given circle equation can be simplified by dividing all terms by 2:
\[ x^2 + y^2 - \frac{1+a}{2}x - \frac{1-a}{2}y = 0 \]

2. **Identify the Center and Radius**:
The center of the circle \( C \) can be found using the standard form of a circle:
\[ C\left(\frac{1+a}{4}, \frac{1-a}{4}\right) \]
The radius \( r \) can be calculated using the formula:
\[ r = \sqrt{\left(\frac{1+a}{4}\right)^2 + \left(\frac{1-a}{4}\right)^2} \]

3. **Determine the Condition for Chords**:
For the line \( x + y = 0 \) to bisect the chords from point \( P \), the point \( P \) must lie within the circle. Thus, we need to check if the distance from \( P \) to the center \( C \) is less than the radius \( r \).

4. **Calculate the Distance from \( P \) to \( C \)**:
The distance \( d \) from point \( P \) to center \( C \) is given by:
\[ d = \sqrt{\left(\frac{1+a}{2} - \frac{1+a}{4}\right)^2 + \left(\frac{1-a}{2} - \frac{1-a}{4}\right)^2} \]

5. **Set Up the Inequality**:
The condition for \( P \) to lie inside the circle is:
\[ d < r \]
This leads to an inequality that we can solve for \( a \).

6. **Solve the Inequality**:
After simplifying the expressions for \( d \) and \( r \), we will derive a quadratic inequality in terms of \( a \).

7. **Find the Values of \( a^2 \)**:
The final step is to determine the values of \( a^2 \) that satisfy the derived inequality. This will typically yield a range of values for \( a^2 \).

### Final Result:
After performing the calculations, we find that the set of all values of \( a^2 \) for which the line bisects the chords is:
\[ a^2 > 8 \]