Multiple choice

In a circle of radius of 3 units, a diameter AB intersects a chord of length 2 units perpendicular at P. If AP > BP, then what is the ratio of AP to BP? 3 इकाई त्रिज्या वाले एक वृत्त में, एक व्यास AB, 2 मीटर लम्बाई की एक जीवा को लम्बवत बिंदु P पर प्रतिच्छेद करता है । यदि AP > BP है, तो AP का BP से क्या अनुपात है?

  1. 3+ 8 :3− 8

  2. 3+ 10 :3− 10

  3. 3+ 3 :3− 3

  4. 3: 3
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A Correct answer
Explanation

In a circle, when a diameter intersects a chord perpendicularly at P: AP × BP = (chord half-length)² = 1² = 1. Also AP + BP = 2r = 6. Solving: If AP = x, BP = 6-x, then x(6-x) = 1, giving x² - 6x + 1 = 0. Solving: x = (6 ± √(36-4))/2 = (6 ± √32)/2 = 3 ± 2√2 = 3 ± √8. Since AP > BP, AP = 3 + √8 and BP = 3 - √8, giving ratio (3+√8):(3-√8).

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