If ` (***) ` is n operation such that a `(***) b = a + b` when ` a gt 0, b gt 0 a (***) b = sqrt( a^(2) + b^2))` for the other value of a and b . The value of `( 8 (***) ( 7 - 13 ) - ( 3 (***) 1))/(( 3 =- 6 ) (***) ( 9 - 5))` is

Options

A. `(1)/(5)`
B. `(4)/(5)`
C. `(6)/(5)`
D. `(2)/(5)`

Answer

To solve the problem step by step, we will use the defined operation \( (***) \) and follow the order of operations.

### Step 1: Identify the operation
The operation \( a \star b \) is defined as follows:
- If \( a > 0 \) and \( b > 0 \), then \( a \star b = a + b \).
- For all other values of \( a \) and \( b \), \( a \star b = \sqrt{a^2 + b^2} \).

### Step 2: Break down the expression
We need to evaluate the expression:
\[ \frac{(8 \star (7 - 13)) - (3 \star 1)}{(3 - 6) \star (9 - 5)} \]

First, simplify the terms inside the parentheses:
- \( 7 - 13 = -6 \)
- \( 9 - 5 = 4 \)

Now, rewrite the expression:
\[ \frac{(8 \star (-6)) - (3 \star 1)}{(-3) \star 4} \]

### Step 3: Evaluate each operation

#### 3.1: Evaluate \( 8 \star (-6) \)
Here, \( a = 8 \) and \( b = -6 \). Since \( b \) is not greater than 0, we use the second definition:
\[ 8 \star (-6) = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

#### 3.2: Evaluate \( 3 \star 1 \)
Here, \( a = 3 \) and \( b = 1 \). Both are greater than 0, so we use the first definition:
\[ 3 \star 1 = 3 + 1 = 4 \]

#### 3.3: Evaluate \( (-3) \star 4 \)
Here, \( a = -3 \) and \( b = 4 \). Since \( a \) is not greater than 0, we use the second definition:
\[ (-3) \star 4 = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Substitute the evaluated values back into the expression
Now we can substitute the values we found:
\[ \frac{(10) - (4)}{(5)} = \frac{6}{5} \]

### Final Result
Thus, the value of the expression is:
\[ \frac{6}{5} \]

### Conclusion
The answer is \( \frac{6}{5} \), which corresponds to option 3.