If `a="max"{(x+2)^(2)+(y-3)^(2)} and b="min"{(x+2)^(2)+(y-3)^(2)}` where x, y satisfying `x^(2)+y^(2)+8x-10y-40=0` then :

Options

A. `a+b=18`
B. `a+b=158`
C. `a-b=4sqrt(2)`
D. `a-b=72sqrt(2)`

Answer

To solve the problem, we need to find the maximum and minimum values of the expression \((x+2)^2 + (y-3)^2\) given the constraint defined by the equation \(x^2 + y^2 + 8x - 10y - 40 = 0\).

### Step-by-Step Solution:

1. **Rewrite the Circle Equation:**
We start with the equation:
\[ x^2 + y^2 + 8x - 10y - 40 = 0 \]
We can complete the square for both \(x\) and \(y\).

- For \(x\):
\[ x^2 + 8x = (x+4)^2 - 16 \]
- For \(y\):
\[ y^2 - 10y = (y-5)^2 - 25 \]

Substituting these back into the equation gives:
\[ (x+4)^2 - 16 + (y-5)^2 - 25 - 40 = 0 \]
Simplifying this, we get:
\[ (x+4)^2 + (y-5)^2 = 81 \]

This represents a circle centered at \((-4, 5)\) with a radius of \(9\).

2. **Identify the Expression to Maximize/Minimize:**
We need to maximize and minimize:
\[ (x+2)^2 + (y-3)^2 \]

3. **Rewrite the Expression:**
We can also rewrite this expression:
\[ (x+2)^2 + (y-3)^2 = \left((x+4) - 2\right)^2 + \left((y-5) + 2\right)^2 \]

4. **Geometric Interpretation:**
The expression \((x+2)^2 + (y-3)^2\) represents the squared distance from the point \((-2, 3)\) to any point \((x, y)\) on the circle centered at \((-4, 5)\).

5. **Calculate the Distance from the Center to the Point:**
The distance from the center of the circle \((-4, 5)\) to the point \((-2, 3)\) is:
\[ d = \sqrt{((-2) - (-4))^2 + (3 - 5)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

6. **Finding Maximum and Minimum Distances:**
- The maximum distance from \((-2, 3)\) to the circle is:
\[ d + r = 2\sqrt{2} + 9 \]
- The minimum distance from \((-2, 3)\) to the circle is:
\[ |d - r| = |2\sqrt{2} - 9| \]

7. **Calculating Maximum and Minimum Values:**
- Maximum value \(a\):
\[ a = (d + r)^2 = (2\sqrt{2} + 9)^2 = (2\sqrt{2})^2 + 2 \cdot (2\sqrt{2}) \cdot 9 + 9^2 = 8 + 36\sqrt{2} + 81 = 89 + 36\sqrt{2} \]
- Minimum value \(b\):
\[ b = (|d - r|)^2 = (9 - 2\sqrt{2})^2 = 81 - 36\sqrt{2} + 8 = 89 - 36\sqrt{2} \]

8. **Final Calculation of \(a + b\):**
\[ a + b = (89 + 36\sqrt{2}) + (89 - 36\sqrt{2}) = 178 \]

Thus, the final answer is:
\[ \boxed{178} \]