Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $x, y$ be real variable satisfy the $x^2 + y^2 + 8x - 10y - 40 = 0$. Let $a = \max\sqrt{(x+2)^2 + (y-3)^2}$ and $b = \min\sqrt{(x+2)^2 + (y-3)^2}$, then:
$a + b = 18$
$a + b = 4\sqrt{2}$
$a - b = 4\sqrt{2}$
$ab = 73$
Step-by-Step Solution
Key Concept: Use Vieta's formulas to find two unknowns from their sum and product by constructing and solving a quadratic equation.
Given the product $ab = 81 - 8 = 73$ and sum $a + b = 18$, solving the quadratic $t^2 - 18t + 73 = 0$ yields $a = 9 + 2\sqrt{2}$ and $b = 9 - 2\sqrt{2}$. The difference is $a - b = 4\sqrt{2}$.
Correct Answer: 1,3,4