In a defined mathematical coding language, the binary operation $a\$b$ is evaluated using the formula $a\$b=a^2-b$. If $a\$b=2$, determine which pair represents valid values for a and b, respectively.

Correct Answer: The correct option representing valid values for $a$ and $b$ respectively is 2, 2.

Step-by-Step Explanation:

We are given the definition of a binary operation:

$a\$b=a^2-b$

We need to find the pair $(a,b)$ from the given choices such that:

$a\$b=2$

Let's evaluate the operation $a^2-b$ for each given pair:

1. For the pair 9, 6 ($a=9,b=6$):

$9^2-6=81-6=75\ne 2$

2. For the pair 8, 1 ($a=8,b=1$):

$8^2-1=64-1=63\ne 2$

3. For the pair 4, 5 ($a=4,b=5$):

$4^2-5=16-5=11\ne 2$

4. For the pair 2, 2 ($a=2,b=2$):

$2^2-2=4-2=2$

This matches the required result of $2$. Therefore, the pair 2, 2 represents valid values for $a$ and $b$.