यदि l, b और p एक आयत की लंबाई, चौड़ाई और परिधि हैं, और b, l और p गुणोत्तर श्रेणी में हैं (क्रम में), तो  (l/b)2 है:

  1. (√3 + 1) : 1
  2. (4 + 2√3) : 1
  3. (2 + 4√3) : 1
  4. 4 : √3

Answer (Detailed Solution Below)

Option 2 : (4 + 2√3) : 1
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Detailed Solution

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दिया गया है:

l, b और p गुणोत्तर श्रेणी में हैं, जहाँ

l = लंबाई, b = चौड़ाई और p = परिधि 

प्रयुक्त अवधारणा:

(a - b)2 = a2 - 2ab + b2

(a + b)2 = a2 + 2ab + b2

आयत की परिधि = 2 × (l + b)

यदि a, b और c गुणोत्तर श्रेणी में हैं (क्रम में) तो,

b2 = ac

स्पष्टीकरण:

प्रश्न के अनुसार

b, l, p गुणोत्तर श्रेणी में हैं

इसलिए, l2 = b × p

⇒ l2 = b × 2(l + b)

⇒ l2 - 2lb = 2b2

⇒ l2 - 2lb + b2 = 2b2 + b2

⇒ (l - b)2 = 3b2

⇒ l - b = √3b

⇒ l = √3b + b

⇒ l = b (√3 + 1)

⇒ l/b = (√3 + 1)/1

⇒ (l/b)2 = (4 + 2√3)/1

∴  l2 : b2 का अनुपात (4 + 2√3) : 1 है।

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