In the xy - plane, the point (0, 2) is the center of a circle that has radius 2. Which of the following is NOT a point on the circle?

Options

A. `(0, 4)`
B. `(-2, 4)`
C. `(-2, 2)`
D. `(0, 0)`

Answer

To determine which point is NOT on the circle with center (0, 2) and radius 2, we will follow these steps:

### Step 1: Write the equation of the circle
The standard equation of a circle with center (a, b) and radius r is given by:

\[ (x - a)^2 + (y - b)^2 = r^2 \]

In this case, the center is (0, 2) and the radius is 2. Thus, we can substitute these values into the equation:

\[ (x - 0)^2 + (y - 2)^2 = 2^2 \]

This simplifies to:

\[ x^2 + (y - 2)^2 = 4 \]

### Step 2: Identify the points to test
Let's assume we have several points to test whether they lie on the circle or not. For example, let's consider the following points:
1. A: (-2, 2)
2. B: (-2, 4)
3. C: (2, 2)
4. D: (0, 0)

### Step 3: Substitute each point into the equation
We will substitute each point into the equation \(x^2 + (y - 2)^2 = 4\) to see if it satisfies the equation.

#### Testing Point A: (-2, 2)
\[ (-2)^2 + (2 - 2)^2 = 4 + 0 = 4 \]
This point satisfies the equation.

#### Testing Point B: (-2, 4)
\[ (-2)^2 + (4 - 2)^2 = 4 + 2^2 = 4 + 4 = 8 \]
This point does NOT satisfy the equation.

#### Testing Point C: (2, 2)
\[ (2)^2 + (2 - 2)^2 = 4 + 0 = 4 \]
This point satisfies the equation.

#### Testing Point D: (0, 0)
\[ (0)^2 + (0 - 2)^2 = 0 + 4 = 4 \]
This point satisfies the equation.

### Conclusion
The point that does NOT lie on the circle is **(-2, 4)**.