बिंदु (2, 3, 4) से रेखा \(\rm \frac{x-0}{1}=\frac{y-0}{0}=\frac{z-0}{0} \ ?\) तक लंबवत दूरी क्या है?

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  1. 6 इकाइयाँ
  2. 5 इकाइयाँ
  3. 3 इकाइयाँ
  4. 2 इकाइयाँ

Answer (Detailed Solution Below)

Option 2 : 5 इकाइयाँ
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संकल्पना:

दो लंबवत रेखाओं का बिंदु गुणनफल शून्य है।

दो बिंदुओं (x1, y1, z1) और (x2, y2, z2) के बीच की दूरी निम्न द्वारा दी जाती है, \(\rm \dfrac{x-0}{1}=\dfrac{y-0}{0}=\dfrac{z-0}{0} =k\)

 

गणना:

माना कि M बिंदु P (2, 3, 4) से खींचे हुए लंब का पाद है

माना कि \(\rm \dfrac{x-0}{1}=\dfrac{y-0}{0}=\dfrac{z-0}{0} =k\)

x = k, y = 0, z = 0

तो M = (k, 0, 0)

अब PM के दिशा अनुपात = (2 - k, 3 - 0, 4 - 0) = (2- k, 3, 4) और दी गई रेखा के दिशा अनुपात 1, 0, 0 हैं।

PM दी गई रेखा के लंबवत है,

(2 - k) (1) + 3(0) + 4 (0) = 0

∴ k = 2

M = (2, 0, 0)

लम्बवत दूरी PM =

\(\rm \sqrt {(2-2)^2+(0-3)^2+(0-4)^2}\\ =\sqrt{9+16}\\ =5\)

 

इसलिए, विकल्प (2) सही है।

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