The geometric mean of the minimum and maximum values of the distance of point (-7, 2) from the points on the circle `x^(2)+y^(2)-10x-14y-51=0` is equal to

Options

A. `2sqrt(11)`
B. 13
C. `5sqrt(5)`
D. 12

Explanation

To solve the problem, we need to find the geometric mean of the minimum and maximum distances from the point (-7, 2) to the points on the circle defined by the equation \(x^2 + y^2 - 10x - 14y - 51 = 0\).

### Step-by-Step Solution:

1. **Rewrite the Circle's Equation**:
The given equation of the circle is:
\[ x^2 + y^2 - 10x - 14y - 51 = 0 \]
We need to convert this into the standard form \((x - h)^2 + (y - k)^2 = r^2\).

2. **Complete the Square**:
- For \(x\):
\[ x^2 - 10x \rightarrow (x - 5)^2 - 25 \]
- For \(y\):
\[ y^2 - 14y \rightarrow (y - 7)^2 - 49 \]
Substituting these into the equation gives:
\[ (x - 5)^2 - 25 + (y - 7)^2 - 49 - 51 = 0 \]
Simplifying:
\[ (x - 5)^2 + (y - 7)^2 - 125 = 0 \]
Thus, the equation of the circle is:
\[ (x - 5)^2 + (y - 7)^2 = 125 \]
This shows that the center of the circle is \((5, 7)\) and the radius \(r\) is \(\sqrt{125} = 5\sqrt{5}\).

3. **Calculate the Distance from the Point to the Center**:
We need to find the distance \(d\) from the point \((-7, 2)\) to the center of the circle \((5, 7)\):
\[ d = \sqrt{(5 - (-7))^2 + (7 - 2)^2} \]
Simplifying:
\[ d = \sqrt{(5 + 7)^2 + (7 - 2)^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

4. **Determine the Maximum and Minimum Distances**:
- The maximum distance \(D_{max}\) from the point to the circle is:
\[ D_{max} = d + r = 13 + 5\sqrt{5} \]
- The minimum distance \(D_{min}\) from the point to the circle is:
\[ D_{min} = d - r = 13 - 5\sqrt{5} \]

5. **Calculate the Geometric Mean**:
The geometric mean \(GM\) of \(D_{max}\) and \(D_{min}\) is given by:
\[ GM = \sqrt{D_{max} \cdot D_{min}} = \sqrt{(13 + 5\sqrt{5})(13 - 5\sqrt{5})} \]
Using the difference of squares:
\[ GM = \sqrt{13^2 - (5\sqrt{5})^2} = \sqrt{169 - 125} = \sqrt{44} = 2\sqrt{11} \]

### Final Answer:
The geometric mean of the minimum and maximum distances is:
\[ \boxed{2\sqrt{11}} \]