Statement 1 the condition on a and b for which two distinct chords of the ellipse `x^(2)/a^(2)+y^(2)/b^(2)=2` passing through (a,-b) are bisected by the line x+y=b is `a^(2)+6ab-7b^(2)gt0`.
Statement 2 Equation of chord of the ellipse `x^(2)/a^(2)+y^(2)/b^(2)=1` whose mid-point `(x_1,y_1)` is `T=S_1`

Answer

To solve the problem, we need to analyze the statements and derive the necessary conditions step by step.

### Step-by-Step Solution

1. **Understanding the Ellipse Equation**:
The equation of the ellipse given is:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \]
This can be rewritten as:
\[ \frac{x^2}{2a^2} + \frac{y^2}{2b^2} = 1 \]
This represents an ellipse centered at the origin with semi-major axis \(\sqrt{2}a\) and semi-minor axis \(\sqrt{2}b\).

2. **Chords through a Point**:
We need to find the condition under which two distinct chords of the ellipse pass through the point \((a, -b)\) and are bisected by the line \(x + y = b\).

3. **Midpoint of the Chord**:
The midpoint of a chord can be represented as \((x_1, y_1)\). For the line \(x + y = b\), we can express \(y_1\) in terms of \(x_1\):
\[ y_1 = b - x_1 \]

4. **Using the Chord Equation**:
The equation of the chord of the ellipse with midpoint \((x_1, y_1)\) can be derived from the standard form of the ellipse:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 2 \]
Substituting \(y = b - x\) into the ellipse equation gives us:
\[ \frac{x^2}{a^2} + \frac{(b - x)^2}{b^2} = 2 \]

5. **Expanding the Equation**:
Expanding the equation:
\[ \frac{x^2}{a^2} + \frac{b^2 - 2bx + x^2}{b^2} = 2 \]
This simplifies to:
\[ \frac{x^2}{a^2} + 1 - \frac{2b}{b^2}x + \frac{x^2}{b^2} = 2 \]
Rearranging gives:
\[ \left(\frac{1}{a^2} + \frac{1}{b^2}\right)x^2 - \frac{2}{b}x + (1 - 2) = 0 \]

6. **Condition for Distinct Chords**:
For the quadratic equation in \(x\) to have two distinct solutions, the discriminant must be greater than zero:
\[ \left(-\frac{2}{b}\right)^2 - 4\left(\frac{1}{a^2} + \frac{1}{b^2}\right)(-1) > 0 \]
Simplifying this gives:
\[ \frac{4}{b^2} + 4\left(\frac{1}{a^2} + \frac{1}{b^2}\right) > 0 \]
This leads to the condition:
\[ a^2 + 6ab - 7b^2 > 0 \]

### Conclusion
Thus, the condition on \(a\) and \(b\) for which two distinct chords of the ellipse pass through the point \((a, -b)\) and are bisected by the line \(x + y = b\) is:
\[ a^2 + 6ab - 7b^2 > 0 \]