Problem: A circle with a circumscribed and an inscribed square centered at the origin 0 of a rectangular coordinate system with positive x and y axes OX and OY, is shown in each figure I to IV below
The inequalities
∣x∣+∣y∣≤2(x2+y2)≤2Max(∣x∣,∣y∣)
are represented geometrically by the figure numbered
Answer Choices:
A. I
B. II
C. III
D. IV
E. None of these
Solution:
Cosider the figure for some positive integer r. Tbe equation of the cuter diamood is ∣x∣+∣y∣=r. Points on or inside the diamond satisfy (1) ∣x∣+∣y∣=r. The equations of the circie is x2+y2=2r2, so points on the circle satisify (2)2(x2+y2)=r.
The equation of the inner square is Max(∣x∣,∣y∣)=2r, so points on or contaide the square satisfy (3)r≤2Max(∣x∣,∣y∣)
Thus from (1),(2), and (3) it is seen that points on the circle antiafy ∣x∣+∣y∣≤r=2(x2+y2)≤2Max(∣x∣,∣y∣).
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