The normal to the circle `x^(2) + y^(2) -2x -2y = 0` passing through (2,2) is

Options

A. x=y
B. `2x + y -6=0`
C. `x + 2y -6= 0`
D. `x + y -4 =0`

Answer

To find the normal to the circle given by the equation \(x^2 + y^2 - 2x - 2y = 0\) that passes through the point (2, 2), we can follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we need to rewrite the equation of the circle in a standard form. We can complete the square for both \(x\) and \(y\).

The given equation is:
\[ x^2 + y^2 - 2x - 2y = 0 \]

Rearranging gives:
\[ x^2 - 2x + y^2 - 2y = 0 \]

Completing the square for \(x\):
\[ x^2 - 2x = (x-1)^2 - 1 \]

Completing the square for \(y\):
\[ y^2 - 2y = (y-1)^2 - 1 \]

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

### Step 2: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center \((h, k) = (1, 1)\)
- Radius \(r = \sqrt{2}\)

### Step 3: Check if the Point Lies on the Circle
Now, we check if the point (2, 2) lies on the circle:
\[ (2-1)^2 + (2-1)^2 = 1^2 + 1^2 = 1 + 1 = 2 \]
Since the left-hand side equals the right-hand side, the point (2, 2) lies on the circle.

### Step 4: Find the Slope of the Radius
Next, we need to find the slope of the radius from the center (1, 1) to the point (2, 2):
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{2 - 1} = \frac{1}{1} = 1 \]

### Step 5: Find the Slope of the Normal
The slope of the normal line is the negative reciprocal of the slope of the radius. Therefore:
\[ \text{slope of normal} = -\frac{1}{1} = -1 \]

### Step 6: Write the Equation of the Normal
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1) = (2, 2)\) and \(m = -1\):
\[ y - 2 = -1(x - 2) \]
Simplifying this:
\[ y - 2 = -x + 2 \]
\[ y = -x + 4 \]

Thus, the equation of the normal to the circle passing through the point (2, 2) is:
\[ y = -x + 4 \]

### Summary of Steps
1. Rewrite the circle's equation in standard form.
2. Identify the center and radius.
3. Check if the given point lies on the circle.
4. Calculate the slope of the radius.
5. Find the slope of the normal.
6. Write the equation of the normal.