If the set of `'K'` for which two distinct chords of the ellipse `(x^(2))/8+(y^(2))/2=1` passing through `(2,-1)` are bisected by the line `x+y=K` is `[a,b]` then `(a+b)` is…………

Explanation

To solve the problem, we need to find the values of \( K \) for which two distinct chords of the ellipse given by

\[ \frac{x^2}{8} + \frac{y^2}{2} = 1 \]

passing through the point \( (2, -1) \) are bisected by the line \( x + y = K \).

### Step 1: Set up the equation of the line
The equation of the line is given as \( x + y = K \). We can express \( y \) in terms of \( x \):

\[ y = K - x \]

### Step 2: Find the coordinates of the midpoint
Let the coordinates of the midpoint of the chord be \( (t, K - t) \).

### Step 3: Use the midpoint formula
The midpoint of a chord that passes through the point \( (2, -1) \) can be expressed as:

\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Given that the chord passes through \( (2, -1) \), we can write the equations for the points on the ellipse in terms of \( t \):

\[ \frac{t}{8} + \frac{K - t}{2} = 1 \]

### Step 4: Substitute into the ellipse equation
Substituting \( y = K - t \) into the ellipse equation gives:

\[ \frac{t^2}{8} + \frac{(K - t)^2}{2} = 1 \]

### Step 5: Expand and simplify the equation
Expanding the equation:

\[ \frac{t^2}{8} + \frac{K^2 - 2Kt + t^2}{2} = 1 \]

Combining terms:

\[ \frac{t^2}{8} + \frac{4K^2 - 8Kt + 4t^2}{8} = 1 \]

This simplifies to:

\[ \frac{5t^2 - 8Kt + 4K^2 - 8}{8} = 0 \]

### Step 6: Form the quadratic equation
Multiplying through by 8 gives:

\[ 5t^2 - 8Kt + (4K^2 - 8) = 0 \]

### Step 7: Condition for distinct chords
For the chords to be distinct, the discriminant of this quadratic must be greater than zero:

\[ D = (-8K)^2 - 4 \cdot 5 \cdot (4K^2 - 8) > 0 \]

Calculating the discriminant:

\[ 64K^2 - 20(4K^2 - 8) > 0 \]

This simplifies to:

\[ 64K^2 - 80K^2 + 160 > 0 \]

\[ -16K^2 + 160 > 0 \]

\[ 16K^2 < 160 \]

\[ K^2 < 10 \]

### Step 8: Find the range for K
Taking the square root gives:

\[ -\sqrt{10} < K < \sqrt{10} \]

### Step 9: Calculate \( a + b \)
Here, \( a = -\sqrt{10} \) and \( b = \sqrt{10} \). Thus,

\[ a + b = -\sqrt{10} + \sqrt{10} = 0 \]

### Final Answer
The value of \( a + b \) is:

\[ \boxed{0} \]