STATEMENT-1 : The equation of chord of circle `x^(2) + y^(2) - 6x + 10y - 9 = 0`, which is be bisected at `(-2, 4)` must be x + y = 2.
and
STATEMENT-2 : The equation of chord with mid-point `(x_(1), y_(1))` to the circle `x^(2) + y^(2) = r^(2)` is `xx_(1) + yy_(1) = x_(1)^(2) + y^(2)`.

Options

A. Statement-1 is true, statement-2 is true, Statement -2 is a correct explanation for Statement -1
B. Statement -1 is true, Statement-2 is true , Statement-2 is NOT a correct explanation for statement-1
C. Statement-1 is true, Statement-2 is False
D. Statement-1 is False, Statement-2 is true

Answer

To solve the problem, we will analyze both statements one by one.

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

**Step 1.1: Rewrite the Circle Equation**
We can rewrite the equation 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 back into the equation:
\[ (x - 3)^2 - 9 + (y + 5)^2 - 25 - 9 = 0 \]
\[ (x - 3)^2 + (y + 5)^2 - 43 = 0 \]
Thus, the equation of the circle becomes:
\[ (x - 3)^2 + (y + 5)^2 = 43 \]
This indicates that the center of the circle is at \((3, -5)\) and the radius is \(\sqrt{43}\).

**Step 1.2: Use the Chord Midpoint Formula**
The midpoint of the chord is given as \((-2, 4)\). The equation of the chord can be derived using the formula:
\[ T = S_1 \]
Where \(T\) is given by:
\[ T: x \cdot x_1 + y \cdot y_1 = r^2 \]
And \(S_1\) is obtained by substituting the midpoint into the circle's equation.

1. Substitute \((-2, 4)\) into the circle's equation:
\[ S_1 = (-2)^2 + (4)^2 - 6(-2) + 10(4) - 9 \]
\[ = 4 + 16 + 12 + 40 - 9 = 63 \]

2. Now, substituting into the \(T\) equation:
\[ T: x \cdot (-2) + y \cdot 4 = 63 \]
This simplifies to:
\[ -2x + 4y = 63 \]
Dividing through by -1 gives:
\[ 2x - 4y = -63 \quad \text{or} \quad x - 2y = -31.5 \]
This does not match \(x + y = 2\).

### Conclusion for Statement 1
Thus, Statement 1 is **false**.

### Step 2: Analyze Statement 2
The statement claims that the equation of a chord with midpoint \((x_1, y_1)\) to the circle \(x^2 + y^2 = r^2\) is:
\[ xx_1 + yy_1 = x_1^2 + y_1^2 \]

**Step 2.1: Verify Statement 2**
Using the same chord midpoint formula \(T = S_1\):
1. For the circle \(x^2 + y^2 = r^2\), we have:
\[ T: xx_1 + yy_1 = r^2 \]
And substituting the midpoint into the circle gives:
\[ S_1: x_1^2 + y_1^2 - r^2 = 0 \]
Thus:
\[ xx_1 + yy_1 = x_1^2 + y_1^2 \]
This confirms the statement is **true**.

### Final Conclusion
- Statement 1 is **false**.
- Statement 2 is **true**.