The farthest distance of the point (1,5) from the circle `(x - 1)^(2) + ( y + 1) ^(2) = 16` is

Options

A. 4
B. 5
C. 6
D. 10

Answer

To find the farthest distance of the point (1, 5) from the circle given by the equation \((x - 1)^2 + (y + 1)^2 = 16\), we will follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle can be rewritten as:
\[ (x - 1)^2 + (y + 1)^2 = 4^2 \]
From this, we can identify:
- The center of the circle \(C\) is at the point \((1, -1)\).
- The radius \(r\) of the circle is \(4\).

### Step 2: Calculate the distance from the point to the center of the circle
We need to find the distance \(PC\) from the point \(P(1, 5)\) to the center \(C(1, -1)\). We can use the distance formula:
\[ PC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \(P\) and \(C\):
\[ PC = \sqrt{(1 - 1)^2 + (5 - (-1))^2} = \sqrt{0^2 + (5 + 1)^2} = \sqrt{0 + 6^2} = \sqrt{36} = 6 \]

### Step 3: Calculate the farthest distance from the point to the circle
The farthest distance from the point \(P\) to the circle is given by the formula:
\[ \text{Farthest distance} = PC + r \]
Substituting the values we have:
\[ \text{Farthest distance} = 6 + 4 = 10 \]

### Final Answer
Thus, the farthest distance of the point (1, 5) from the circle is:
\[ \boxed{10} \]