`(1, 2sqrt2)`is a point on circle, `x^2 + y^2 = 9`. Which of the following is not the point on the circle at 2 units distance from `(1, 2sqrt2)`?

Options

A. `(-1,2sqrt(2))`
B. `(2sqrt(2),1)`
C. `((23)/(9),(10sqrt(2))/(9))`
D. None of these

Explanation

To solve the problem, we need to determine which of the given points is not on the circle defined by the equation \( x^2 + y^2 = 9 \) and is also at a distance of 2 units from the point \( (1, 2\sqrt{2}) \).

### Step 1: Find the distance formula
We will use the distance formula to find points that are 2 units away from \( (1, 2\sqrt{2}) \). The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, we want \( d = 2 \) and \( (x_1, y_1) = (1, 2\sqrt{2}) \).

### Step 2: Set up the equation
Substituting into the distance formula gives:
\[ \sqrt{(x - 1)^2 + (y - 2\sqrt{2})^2} = 2 \]
Squaring both sides, we get:
\[ (x - 1)^2 + (y - 2\sqrt{2})^2 = 4 \]

### Step 3: Expand the equation
Expanding the equation:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ (y - 2\sqrt{2})^2 = y^2 - 4\sqrt{2}y + 8 \]
Combining these, we have:
\[ x^2 - 2x + 1 + y^2 - 4\sqrt{2}y + 8 = 4 \]
Simplifying this gives:
\[ x^2 + y^2 - 2x - 4\sqrt{2}y + 5 = 0 \]

### Step 4: Substitute the circle equation
Now, we know that \( x^2 + y^2 = 9 \) from the circle equation. We can substitute this into our equation:
\[ 9 - 2x - 4\sqrt{2}y + 5 = 0 \]
This simplifies to:
\[ -2x - 4\sqrt{2}y + 14 = 0 \]
Rearranging gives:
\[ 2x + 4\sqrt{2}y = 14 \]
Dividing through by 2:
\[ x + 2\sqrt{2}y = 7 \]

### Step 5: Identify the points
Now we need to find points that satisfy both the circle equation \( x^2 + y^2 = 9 \) and the line equation \( x + 2\sqrt{2}y = 7 \).

### Step 6: Solve the equations
We can express \( x \) in terms of \( y \) from the line equation:
\[ x = 7 - 2\sqrt{2}y \]
Substituting this into the circle equation:
\[ (7 - 2\sqrt{2}y)^2 + y^2 = 9 \]
Expanding and simplifying will yield the possible values for \( y \), and subsequently for \( x \).

### Step 7: Check the options
Once we find the points that satisfy both equations, we can check which of the given options is not one of these points.