Question:

Step 1: Identify the given sum and product of $a$ and $b$. The problem implicitly provides the sum and product of $a$ and $b$, which are derived from the geometric interpretation of the problem (distance from a point to points on a circle). The sum of $a$ and $b$ is given as: $$a + b = 18$$ The product of $a$ and $b$ is given as: $$ab = 81 - 8 = 73$$ Step 2: Form a quadratic equation with $a$ and $b$ as its roots. If $a$ and $b$ are the roots of a quadratic equation, the equation can be expressed in the form $t^2 - (a+b)t + ab = 0$. Substituting the values from Step 1: $$t^2 - 18t + 73 = 0$$ Step 3: Solve the quadratic equation to find the values of $a$ and $b$. We use the quadratic formula $t = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$ for the equation $t^2 - 18t + 73 = 0$, where $A=1$, $B=-18$, and $C=73$. First, calculate the discriminant $\Delta = B^2 - 4AC$: $$\Delta = (-18)^2 - 4(1)(73) = 324 - 292 = 32$$ Now, substitute the values into the quadratic formula: $$t = \frac{-(-18) \pm \sqrt{32}}{2(1)}$$ $$t = \frac{18 \pm \sqrt{16 \times 2}}{2}$$ $$t = \frac{18 \pm 4\sqrt{2}}{2}$$ $$t = 9 \pm 2\sqrt{2}$$ Since $a$ is defined as the maximum value and $b$ as the minimum value, we assign: $$a = 9 + 2\sqrt{2}$$ $$b = 9 - 2\sqrt{2}$$ Step 4: Calculate the difference $a-b$. Now, we subtract $b$ from $a$: $$a - b = (9 + 2\sqrt{2}) - (9 - 2\sqrt{2})$$ $$a - b = 9 + 2\sqrt{2} - 9 + 2\sqrt{2}$$ $$a - b = 4\sqrt{2}$$ Step 5: Verify the given options. Based on our calculations: 1. $a + b = (9 + 2\sqrt{2}) + (9 - 2\sqrt{2}) = 18$. This matches Option 1. 2. $a - b = 4\sqrt{2}$. This matches Option 3. 3. $ab = (9 + 2\sqrt{2})(9 - 2\sqrt{2}) = 9^2 - (2\sqrt{2})^2 = 81 - (4 \times 2) = 81 - 8 = 73$. This matches Option 4. Option 2 states $a+b = 4\sqrt{2}$, which is incorrect as $a+b=18$. The final answer is that options 1, 3, and 4 are correct.