A circle center (3, 8) contains the point (2, -1). Which of the following is also a point on the circle ?

Options

A. (1, -10)
B. (4, 17)
C. (5, -9)
D. (7, 15)

Explanation

To solve the problem step by step, we need to determine which of the given points lies on the circle centered at (3, 8) that contains the point (2, -1).

### Step 1: Find the radius of the circle
The radius of the circle can be calculated using the distance formula between the center of the circle (3, 8) and the point (2, -1).

The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \((x_1, y_1) = (3, 8)\) and \((x_2, y_2) = (2, -1)\).

Calculating the distance:
\[ d = \sqrt{(2 - 3)^2 + (-1 - 8)^2} \]
\[ = \sqrt{(-1)^2 + (-9)^2} \]
\[ = \sqrt{1 + 81} \]
\[ = \sqrt{82} \]

Thus, the radius \(r\) of the circle is \(\sqrt{82}\).

### Step 2: Write the equation of the circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 3\), \(k = 8\), and \(r = \sqrt{82}\):
\[ (x - 3)^2 + (y - 8)^2 = 82 \]

### Step 3: Check each option to see if it satisfies the equation
We need to check which of the given points satisfies the equation \((x - 3)^2 + (y - 8)^2 = 82\).

Assuming the options are:
- A: (1, -10)
- B: (4, 17)
- C: (5, -9)
- D: (7, 15)

#### Check Option A: (1, -10)
\[ (1 - 3)^2 + (-10 - 8)^2 = (-2)^2 + (-18)^2 = 4 + 324 = 328 \quad \text{(not equal to 82)} \]

#### Check Option B: (4, 17)
\[ (4 - 3)^2 + (17 - 8)^2 = (1)^2 + (9)^2 = 1 + 81 = 82 \quad \text{(equal to 82)} \]

#### Check Option C: (5, -9)
\[ (5 - 3)^2 + (-9 - 8)^2 = (2)^2 + (-17)^2 = 4 + 289 = 293 \quad \text{(not equal to 82)} \]

#### Check Option D: (7, 15)
\[ (7 - 3)^2 + (15 - 8)^2 = (4)^2 + (7)^2 = 16 + 49 = 65 \quad \text{(not equal to 82)} \]

### Conclusion
The only point that satisfies the equation of the circle is **Option B: (4, 17)**.