A Point Spiralling Inward
Difficulty: Medium · Topics: Calculus, Series, Geometric Series
Ichkariga buraluvchi nuqta
A point P starts at the origin O and travels through the points P 1, P 2, P 3, … without end. The first step runs along the positive x-axis with overlineOP 1=1; every step after that turns 90^° anticlockwise from the one before it and is 3/4 as long, so that overlineP 1P 2=3/4 overlineOP 1 runs in the positive y-direction, overlineP 2P 3=3/4 overlineP 1P 2 runs in the negative x-direction, and so on.
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The points P n approach a point (α,\ β). Find the value of (α)/(β).
P nuqta koordinatalar boshi O dan boshlab P 1, P 2, P 3, … nuqtalar orqali cheksiz harakatlanadi. Birinchi qadam x o‘qining musbat yo‘nalishi bo‘ylab va overlineOP 1=1; keyingi har bir qadam oldingisidan soat mili yo‘nalishiga teskari 90^° buriladi va uning 3/4 qismiga teng bo‘ladi, ya’ni overlineP 1P 2=3/4 overlineOP 1 — y ning musbat yo‘nalishida, overlineP 2P 3=3/4 overlineP 1P 2 — x ning manfiy yo‘nalishida va hokazo.
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P n nuqtalar (α,\ β) nuqtaga yaqinlashadi. (α)/(β) ning qiymatini toping.