The shortest distance of the point (8,1) from the circle `(x + 2)^(2) + (y - 1)^(2)` = 25 is

Options

A. 15
B. 10
C. 5
D. 6

Answer

To find the shortest distance from the point (8, 1) to the circle given by the equation \((x + 2)^2 + (y - 1)^2 = 25\), we can follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

From the equation \((x + 2)^2 + (y - 1)^2 = 25\):
- The center \((h, k)\) is \((-2, 1)\).
- The radius \(r\) is \(\sqrt{25} = 5\).

### Step 2: Calculate the distance from the point to the center of the circle
We need to find the distance \(d\) from the point \((8, 1)\) to the center of the circle \((-2, 1)\) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(8 - (-2))^2 + (1 - 1)^2} = \sqrt{(8 + 2)^2 + 0^2} = \sqrt{10^2} = 10 \]

### Step 3: Calculate the shortest distance from the point to the circle
The shortest distance from the point to the circle is given by:
\[ \text{Shortest Distance} = d - r \]
Substituting the values we found:
\[ \text{Shortest Distance} = 10 - 5 = 5 \]

### Final Answer
The shortest distance from the point (8, 1) to the circle is **5 units**.

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