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 given problem, we will analyze both statements step by step.

### Step 1: Analyze the Circle Equation
The equation of the circle is given as:
\[ x^2 + y^2 - 6x + 10y - 9 = 0 \]

We can rewrite this equation in the standard form by completing the square.

### Step 2: Completing the Square
1. For the \(x\) terms:
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]
2. For the \(y\) terms:
\[ y^2 + 10y \rightarrow (y + 5)^2 - 25 \]

Now substituting these back into the equation:
\[ (x - 3)^2 - 9 + (y + 5)^2 - 25 - 9 = 0 \]
\[ (x - 3)^2 + (y + 5)^2 - 43 = 0 \]
\[ (x - 3)^2 + (y + 5)^2 = 43 \]

Thus, the center of the circle is \((3, -5)\) and the radius is \(\sqrt{43}\).

### Step 3: Equation of the Chord
The chord of the circle that is bisected at the point \((-2, 4)\) can be found using the formula:
\[ T = S_1 \]
Where \(T\) is the equation of the chord and \(S\) is the equation of the circle.

### Step 4: Find \(S_1\)
Substituting the midpoint \((-2, 4)\) into the circle equation:
\[ S_1 = (-2)^2 + (4)^2 - 6(-2) + 10(4) - 9 \]
Calculating this:
\[ S_1 = 4 + 16 + 12 + 40 - 9 = 63 \]

### Step 5: Equation of the Chord
Using the formula \(T = S_1\):
\[ T = x(-2) + y(4) - 63 = 0 \]
This simplifies to:
\[ -2x + 4y - 63 = 0 \quad \text{or} \quad 2x - 4y + 63 = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives:
\[ 2x - 4y = -63 \quad \text{or} \quad x + 2y = -31.5 \]

### Step 7: Conclusion on Statement 1
The derived equation does not match \(x + y - 2 = 0\). Therefore, **Statement 1 is FALSE**.

### Step 8: Conclusion on Statement 2
Statement 2 is a general property of conics and is TRUE.

### Final Answer
- Statement 1: FALSE
- Statement 2: TRUE