Question
Download Solution PDFएक त्रिभुज है, जो B पर समकोणीय है। यदि त्रिभुज के शीर्ष A(k,1,−1), B(2k,0,2) और हैं, तो k का मान क्या है?
Answer (Detailed Solution Below)
Detailed Solution
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दिया गया है,
बिंदु A(k, 1, -1), B(2k, 0, 2), और C(2 + 2k, k, 1) त्रिभुज के शीर्ष हैं।
सदिश AB और BC इस प्रकार दिए गए हैं:
\( \overrightarrow{AB} = B - A = (2k - k, 0 - 1, 2 - (-1)) = (k, -1, 3) \)
\( \overrightarrow{BC} = C - B = (2 + 2k - 2k, k - 0, 1 - 2) = (2, k, -1) \)
\( \overrightarrow{AB} \) और \( \overrightarrow{BC} \) का अदिश गुणनफल है:
\( \overrightarrow{AB} \cdot \overrightarrow{BC} = (k)(2) + (-1)(k) + (3)(-1) = 2k - k - 3 = k - 3 \)
चूँकि सदिश लंबवत हैं, इसलिए उनका अदिश गुणनफल शून्य होना चाहिए:
\( k - 3 = 0 \)
इस प्रकार, \( k = 3 \).
∴ k का मान \(3\) है।
इसलिए, सही उत्तर विकल्प 4 है।
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