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=178`
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 \( (x+2)^2 + (y-3)^2 \) under the constraint given by the circle equation \( x^2 + y^2 + 8x - 10y - 40 = 0 \).

### Step-by-Step Solution:

1. **Rewrite the Circle Equation**:
We start with the equation of the circle:
\[ x^2 + y^2 + 8x - 10y - 40 = 0 \]
We can complete the square for both \( x \) and \( y \):
\[ (x^2 + 8x) + (y^2 - 10y) = 40 \]
Completing the square:
\[ (x + 4)^2 - 16 + (y - 5)^2 - 25 = 40 \]
This simplifies to:
\[ (x + 4)^2 + (y - 5)^2 = 81 \]
Thus, the center of the circle is \( (-4, 5) \) and the radius is \( 9 \).

2. **Identify the Expression**:
We need to maximize and minimize:
\[ (x + 2)^2 + (y - 3)^2 \]
This expression represents the square of the distance from the point \( (-2, 3) \) to any point \( (x, y) \) on the circle.

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

4. **Determine Maximum and Minimum Distances**:
The minimum distance from the point \( (-2, 3) \) to the circle is:
\[ \text{Minimum distance} = d - \text{radius} = 2\sqrt{2} - 9 \]
The maximum distance from the point \( (-2, 3) \) to the circle is:
\[ \text{Maximum distance} = d + \text{radius} = 2\sqrt{2} + 9 \]

5. **Calculate \( a \) and \( b \)**:
Since \( a \) is the maximum and \( b \) is the minimum of the expression \( (x + 2)^2 + (y - 3)^2 \):
\[ a = (d + 9)^2 = (2\sqrt{2} + 9)^2 \]
\[ b = (d - 9)^2 = (2\sqrt{2} - 9)^2 \]

6. **Expand \( a \) and \( b \)**:
Expanding \( a \):
\[ a = (2\sqrt{2} + 9)^2 = (2\sqrt{2})^2 + 2 \cdot 2\sqrt{2} \cdot 9 + 9^2 = 8 + 36\sqrt{2} + 81 = 89 + 36\sqrt{2} \]
Expanding \( b \):
\[ b = (2\sqrt{2} - 9)^2 = (2\sqrt{2})^2 - 2 \cdot 2\sqrt{2} \cdot 9 + 9^2 = 8 - 36\sqrt{2} + 81 = 89 - 36\sqrt{2} \]

7. **Final Values of \( a \) and \( b \)**:
Thus, we have:
\[ a = 89 + 36\sqrt{2}, \quad b = 89 - 36\sqrt{2} \]