Problem 16 · 2010 Math Kangaroo
Stretch
Algebra & Patterns
caseworksequence-of-figures
Which of the following graphs represents the solution set of \((x-|x|)^2 + (y-|y|)^2 = 4\)?
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Answer: A
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Hint 1 of 2
The value of x − |x| depends on whether x is negative.
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Hint 2 of 2
Split the plane into the four sign-quadrants and simplify in each.
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Approach: case-split on the signs of x and y
- If a coordinate is non-negative, t−|t| is 0; if it is negative, t−|t| = 2t.
- First quadrant gives 0 = 4 (nothing); the second and fourth quadrants give the rays x = −1 (for y ≥ 0, pointing up) and y = −1 (for x ≥ 0, pointing right).
- The third quadrant gives 4x² + 4y² = 4, a quarter circle x² + y² = 1 joining (−1, 0) to (0, −1).
- The quarter arc together with the upward ray at x = −1 and the rightward ray at y = −1 matches graph A.
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