Find the maximum length of a line segment enclosed in a given area

Question

$A = \{ (x, y) : x = u + v , y = v , (u^2) + (v^2) \le 1 \}$ . Then what is the maximum length of a line segment enclosed in this area?

My friend suggested the answer $\sqrt{5}$, but I think it should be greater.

Answer 1

• $x = u+v$
• $y = v$
• $u^2+v^2 \le 1$

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• $u^2+v^2 \le 1 \implies u,v$ are inside the unit circle, therefore:

  • $-1 \leq u \leq 1$
  • $-1 \leq v \leq 1$
  • $-\sqrt{2} \leq u+v \leq \sqrt{2}$

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• Therefore:

  • $-\sqrt{2} \leq x \leq \sqrt{2}$
  • $-1 \leq y \leq 1$

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The longest line within this rectangle is the diagonal of course, which is equal to:

$$\sqrt{(2\sqrt{2})^2+2^2} = \sqrt{12}$$