If $a\%b=a^2-b^2$ and $a\%b=63$, find a and b.

The correct option is 8, 1.

Let us understand the operation defined in the problem step-by-step.

The problem defines a special binary operation between two numbers $a$ and $b$ as:

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

We are given that:

$a\%b=63$

Substituting the definition into the given equation, we get:

$a^2-b^2=63$

Now, let us test the given options to find which pair of values for $a$ and $b$ satisfies this condition.

Checking Option 1: (9, 6)

Substitute $a=9$ and $b=6$:

$9^2-6^2=81-36=45$

Since $45\ne 63$, this option is incorrect.

Checking Option 2: (10, 7)

Substitute $a=10$ and $b=7$:

${10}^2-7^2=100-49=51$

Since $51\ne 63$, this option is incorrect.

Checking Option 3: (8, 1)

Substitute $a=8$ and $b=1$:

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

Since $63=63$, this option satisfies the given equation.

Checking Option 4: (7, 4)

Substitute $a=7$ and $b=4$:

$7^2-4^2=49-16=33$

Since $33\ne 63$, this option is incorrect.

Therefore, the values of $a$ and $b$ are 8 and 1 respectively.