 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/5/a/solution

The absolute value is nonnegative, so a real point must satisfy

1 - x^{2} \geq 0

, or

- 1 \leq x \leq 1

. On that interval the two branches are

y = 1 - x^{2} or y = x^{2} - 1, - 1 \leq x \leq 1.

They form a closed lens symmetric about both coordinate axes, meeting at

(\pm 1, 0)

and reaching

(0, \pm 1)

. The parabolas outside the stated interval do not belong to the original locus.

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