Statement -1 : The equation of chord of the circel `x ^(2) + y ^(2) - 6x + 10y - 9=0,` which is bisected at the point `(-2,4)` must be `x + y - 2=0.`
Statement -2 : In notations the equation of chord of the circle `S =0` bisected at `x _(1), y _(1)` must be `T = S_(1).`

Answer

To solve the problem, we need to analyze both statements regarding the chord of the circle given by the equation \( x^2 + y^2 - 6x + 10y - 9 = 0 \) and the point at which the chord is bisected, which is \((-2, 4)\).

### Step 1: Rewrite the Circle Equation
The given equation of the circle is:
\[ x^2 + y^2 - 6x + 10y - 9 = 0 \]
We can rewrite it in standard form by completing the square.

1. For \(x\):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
2. For \(y\):
\[ y^2 + 10y = (y + 5)^2 - 25 \]

Substituting these into the equation gives:
\[ (x - 3)^2 - 9 + (y + 5)^2 - 25 - 9 = 0 \]
Simplifying this, we have:
\[ (x - 3)^2 + (y + 5)^2 - 43 = 0 \]
Thus, the standard form of the circle is:
\[ (x - 3)^2 + (y + 5)^2 = 43 \]
This indicates that the center of the circle is \((3, -5)\) and the radius is \(\sqrt{43}\).

### Step 2: Use the Chord Bisector Formula
The equation of the chord of a circle that is bisected at a point \((x_1, y_1)\) can be expressed as:
\[ T = S_1 \]
where \(T\) is the equation of the chord and \(S\) is the equation of the circle.

1. The equation of the circle \(S\) is:
\[ S = x^2 + y^2 - 6x + 10y - 9 \]
2. For the point \((-2, 4)\), we calculate \(S_1\):
\[ S_1 = (-2)^2 + (4)^2 - 6(-2) + 10(4) - 9 \]
\[ = 4 + 16 + 12 + 40 - 9 = 63 \]

### Step 3: Substitute into the Chord Equation
Using the point \((-2, 4)\) in the chord equation \(T\):
\[ T = x(-2) + y(4) - (S_1) = 0 \]
This gives:
\[ -2x + 4y - 63 = 0 \]
Rearranging this, we can express it as:
\[ 2x - 4y + 63 = 0 \]
or simplifying further, we can divide by 2:
\[ x - 2y + 31.5 = 0 \]

### Step 4: Compare with the Given Statement
The statement claims that the equation of the chord is \(x + y - 2 = 0\). However, our derived equation does not match this. Therefore, **Statement 1 is false**.

### Step 5: Validate Statement 2
Statement 2 states that the equation of the chord of the circle \(S = 0\) bisected at \((x_1, y_1)\) must be \(T = S_1\). This is a standard result in coordinate geometry, and we have verified it through our calculations. Thus, **Statement 2 is true**.

### Conclusion
- **Statement 1**: False
- **Statement 2**: True