A circle in the xy-plane is defined by the equation `(x -4) ^(2) + (y + 2)^(2) =100.` Which of the following points is located on the circumference of the circle ?

Options

A. `(-3,5)`
B. `(0,9)`
C. `(4, -2)`
D. `(4,8)`

Answer

To determine which of the given points is located on the circumference of the circle defined by the equation \((x - 4)^2 + (y + 2)^2 = 100\), we will substitute the coordinates of each point into the equation and check if the equation holds true.

### Step-by-Step Solution:

1. **Identify the Circle's Equation**:
The equation of the circle is given as:
\[ (x - 4)^2 + (y + 2)^2 = 100 \]

2. **Substitute the First Point**:
Let's check the point \((-3, 5)\):
\[ x = -3, \quad y = 5 \]
Substituting into the equation:
\[ (-3 - 4)^2 + (5 + 2)^2 = (-7)^2 + (7)^2 = 49 + 49 = 98 \quad (\text{not equal to } 100) \]

3. **Substitute the Second Point**:
Now check the point \((0, 9)\):
\[ x = 0, \quad y = 9 \]
Substituting into the equation:
\[ (0 - 4)^2 + (9 + 2)^2 = (-4)^2 + (11)^2 = 16 + 121 = 137 \quad (\text{not equal to } 100) \]

4. **Substitute the Third Point**:
Next, check the point \((4, -2)\):
\[ x = 4, \quad y = -2 \]
Substituting into the equation:
\[ (4 - 4)^2 + (-2 + 2)^2 = (0)^2 + (0)^2 = 0 \quad (\text{not equal to } 100) \]

5. **Substitute the Fourth Point**:
Finally, check the point \((4, 8)\):
\[ x = 4, \quad y = 8 \]
Substituting into the equation:
\[ (4 - 4)^2 + (8 + 2)^2 = (0)^2 + (10)^2 = 0 + 100 = 100 \quad (\text{equal to } 100) \]

6. **Conclusion**:
The point \((4, 8)\) satisfies the equation of the circle, thus it is located on the circumference of the circle.

### Final Answer:
The point located on the circumference of the circle is \((4, 8)\).