2002 AMC 12B Problems/Problem 25

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Problem

Let $f(x) = x^2 + 6x + 1$ , and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that $f(x) + f(y) \le 0 \qquad \text{and} \qquad f(x)-f(y) \le 0$ The area of $R$ is closest to

$\mathrm{(A)}\ 21\qquad\mathrm{(B)}\ 22\qquad\mathrm{(C)}\ 23\qquad\mathrm{(D)}\ 24\qquad\mathrm{(E)}\ 25$

Solution

The first condition gives us that $x^2 + 6x + 1 + y^2 + 6y + 1 \le 0 \Longrightarrow (x+3)^2 + (y+3)^2 \le 16$

which is a circle centered at $(-3,-3)$ with radius $4$ . The second condition gives us that

$x^2 + 6x + 1 - y^2 - 6y - 1 \le 0 \Longrightarrow (x^2 - y^2) + 6(x-y) \le 0 \Longrightarrow (x-y)(x+y+6) \le 0$

Thus either

$x - y \ge 0,\quad x+y+6 \ge 0$

or

$x - y \le 0,\quad x+y+6 \le 0$

Each of those lines passes through $(-3,-3)$ and has slope $\pm 1$ , as shown above. Therefore, the area of $R$ is half of the area of the circle, which is $\frac{1}{2} (\pi \cdot 4^2) = 8\pi \approx 25 \Rightarrow \mathrm{(E)}$ .

2002 AMC 12B (Problems)
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2002 AMC 12B Problems/Problem 25

From AoPSWiki

Problem

Solution

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