Statement-1: The lines `y=mx+1-m` for all values of m is a normal to the circle `x^(2)+y^(2)-2x-2y=0`.
Statement-2: The line L passes through the centre of the circle.

Options

A. Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
B. Statement-1 is true, statement-2 is true and statement-2 is not the 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 need to analyze both statements given in the question.

### Step 1: Identify the Circle's Center
The equation of the circle is given as:
\[ x^2 + y^2 - 2x - 2y = 0 \]

We can rewrite this in the standard form of a circle:
\[ (x^2 - 2x) + (y^2 - 2y) = 0 \]

To complete the square:
- For \(x^2 - 2x\), we add and subtract \(1\) (which is \((\frac{2}{2})^2\)):
\[ (x - 1)^2 - 1 \]
- For \(y^2 - 2y\), we add and subtract \(1\):
\[ (y - 1)^2 - 1 \]

Putting it all together:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 0 \]
\[ (x - 1)^2 + (y - 1)^2 = 2 \]

From this, we can see that the center of the circle is at:
\[ (h, k) = (1, 1) \]

### Step 2: Analyze the Line
The line is given by:
\[ y = mx + 1 - m \]

We can rearrange this to:
\[ y - mx = 1 - m \]
This is in the slope-intercept form \(y = mx + c\).

### Step 3: Check if the Line Passes Through the Center
To check if this line passes through the center of the circle \((1, 1)\), we substitute \(x = 1\) and \(y = 1\) into the line equation:
\[ 1 = m(1) + 1 - m \]
This simplifies to:
\[ 1 = m + 1 - m \]
\[ 1 = 1 \]

This is true for all values of \(m\), which means the line passes through the center of the circle for all values of \(m\).

### Step 4: Determine if the Line is Normal to the Circle
Since the line passes through the center of the circle, it can be concluded that it is normal to the circle. A normal line to a circle at any point on the circle passes through the center of the circle.

### Conclusion
Both statements are correct:
- Statement 1: The line is normal to the circle.
- Statement 2: The line passes through the center of the circle.

Moreover, Statement 2 provides a correct explanation for Statement 1, as all normals to a circle pass through its center.

### Final Answer
Both statements are true, and Statement 2 is a correct explanation for Statement 1.

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