If `5"@" 6 = 61 and 8 "@ "10 = 164`, then the value of `8 "@" 9 = ?`
यदि `5 "@" 6 = 61 `और `8 "@ "10 = 164`, तो `8 "@" 9` का मान क्या होगा?

Options

A. 145
B. 144
C. 142
D. 164

Explanation

To solve the problem, we need to identify the pattern used in the given operations.

We have the following equations:
1. \( 5 "@" 6 = 61 \)
2. \( 8 "@" 10 = 164 \)

From the video transcript, we can deduce that the operation "@" is defined as follows:
\[ a "@" b = a^2 + b^2 \]

Let's verify this with the given equations:

### Step 1: Verify the first equation
For \( 5 "@" 6 \):
- Calculate \( 5^2 \) and \( 6^2 \):
\[ 5^2 = 25 \]
\[ 6^2 = 36 \]
- Now add these results together:
\[ 5 "@" 6 = 25 + 36 = 61 \]
This confirms that the operation is correct for the first equation.

### Step 2: Verify the second equation
For \( 8 "@" 10 \):
- Calculate \( 8^2 \) and \( 10^2 \):
\[ 8^2 = 64 \]
\[ 10^2 = 100 \]
- Now add these results together:
\[ 8 "@" 10 = 64 + 100 = 164 \]
This confirms that the operation is correct for the second equation.

### Step 3: Calculate \( 8 "@" 9 \)
Now we need to find the value of \( 8 "@" 9 \):
- Calculate \( 8^2 \) and \( 9^2 \):
\[ 8^2 = 64 \]
\[ 9^2 = 81 \]
- Now add these results together:
\[ 8 "@" 9 = 64 + 81 = 145 \]

### Final Answer
Thus, the value of \( 8 "@" 9 \) is \( \boxed{145} \).