Question
Download Solution PDFबिंदु (3, 4, 5) से रेखा \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) तक खींचे गए लंब की लंबाई क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
रेखा \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) और बिंदु (3, 4, 5)
अवधारणा:
यदि दो सदिश एक दूसरे के लंबवत हैं तो दोनों का बिंदु गुणनफल शून्य है।
गणना:
माना बिंदु A = (3, 4, 5)
और \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}=k\)
फिर रेखा पर बिंदु B = (k, 2k + 1, 3k + 2) है।
अब, रेखा AB = B - A = (k - 3, 2k - 3, 3k - 3)
दी गई रेखा (1, 2, 3) के DR
हम जानते हैं कि रेखा AB दी गई रेखा पर लंबवत है
तब
(k - 3, 2k - 3, 3k - 3) ⋅ (1, 2, 3) = 0
⇒ k - 3 +4k - 6 + 9k - 9 = 0
⇒ 14k = 18 ⇒ \(k=\frac{9}{7}\)
फिर रेखा \(\rm AB=\left(\frac{-12}{7},\frac{-3}{7},\frac{-6}{7}\right)\)
दी गई रेखा पर बिंदु A से लंबवत लंबाई AB का परिमाण है।
\(\rm |AB|=\left|\frac{-12}{7},\frac{-3}{7},\frac{-6}{7}\right|\)
\(\rm = \sqrt{\left(\frac{-12}{7}\right)^2+\left(\frac{-3}{7}\right)^2+\left(\frac{-6}{7}\right)^2}\)
\(\rm = \sqrt{\left(\frac{189}{49}\right)}\)
\(\rm = \frac{3\sqrt{21}}{7}\)
अतः विकल्प (2) सही है।
Last updated on Jun 17, 2025
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