The angle between `x^(2)+y^(2)-2x-2y+1=0` and line `y=lambda x + 1-lambda`, is

Options

A. `0^(@)`
B. `45^(@)`
C. `30^(@)`
D. `90^(@)`

Explanation

To find the angle between the given circle and the line, we will follow these steps:

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

### Step 2: Rewrite the Circle Equation in Standard Form
To rewrite the equation in standard form, we will complete the square for both \(x\) and \(y\).

1. Rearranging the terms:
\[ (x^2 - 2x) + (y^2 - 2y) + 1 = 0 \]

2. Completing the square:
- For \(x^2 - 2x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
- For \(y^2 - 2y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

3. Substitute back into the equation:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 + 1 = 0 \]
\[ (x - 1)^2 + (y - 1)^2 - 1 = 0 \]
\[ (x - 1)^2 + (y - 1)^2 = 1 \]

This shows that the circle is centered at \((1, 1)\) with a radius of \(1\).

### Step 3: Identify the Line Equation
The line is given as:
\[ y = \lambda x + 1 - \lambda \]

### Step 4: Determine the Center of the Circle
The center of the circle is at the point \((1, 1)\).

### Step 5: Check if the Line Passes Through the Center
To check if the line passes through the center of the circle, we substitute \(x = 1\) into the line equation:
\[ y = \lambda(1) + 1 - \lambda = 1 \]

Since the line passes through the point \((1, 1)\), we conclude that the line intersects the circle at its center.

### Step 6: Determine the Angle Between the Circle and the Line
The angle between a line and a circle at the point of intersection (in this case, the center) is defined as the angle made by the portion of the line that is intercepted by the circle. Since the line passes through the center, the angle formed is a right angle.

Thus, the angle between the circle and the line is:
\[ \text{Angle} = 90^\circ \]

### Conclusion
The angle between the circle and the line is \(90^\circ\).

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