मूल से तल 4x - 2y + 3z - 6 = 0 तक खींचे गए लंब के पाद के सह-निर्देशांक ज्ञात कीजिए। 

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  1. \(\frac{12}{\sqrt{45}},-\frac{18}{\sqrt{45}}, \frac{24}{\sqrt{45}}\)
  2. \(\frac{24}{45},-\frac{18}{45}, \frac{12}{45}\)
  3. \(\frac{24}{29},-\frac{12}{29}, \frac{18}{29}\)
  4. \(\frac{18}{\sqrt{29}},-\frac{12}{\sqrt{29}}, \frac{24}{\sqrt{29}}\)

Answer (Detailed Solution Below)

Option 3 : \(\frac{24}{29},-\frac{12}{29}, \frac{18}{29}\)
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दिया गया:

तल : 4x - 2y + 3z - 6 = 0

अवधारणा:

समीकरण ax + by + cz = d(a, b, c) वाले तल के लिए तल के अभिलंब के दिशा अनुपात हैं।

(x1, y1, z1) से गुजरने वाली रेखा का समीकरण और कार्तीय रूप में दिशा अनुपात a, b, c है:

\(\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}= \frac{z-z_{1}}{c}\)

गणना:

समतल के लम्बवत् दिशा अनुपात 4 , -2 , और 3 हैं।

मूल (0, 0, 0) और 4, -2, 3 दिशा अनुपात से गुजरने वाली रेखा का समीकरण है:

\(\frac{x-0}{4}=\frac{y-0}{-2}= \frac{z-0}{3} = λ\)

⇒ x = 4λ , y = -2 λ और z = 3λ

उपरोक्त निर्देशांकों के साथ तल का सन्तोषजनक समीकरण :

⇒ 4(4λ) - 2(-2λ) + 3(3λ) - 6 = 0

⇒ λ = 6/29

∴ लंब का पाद,

⇒ x = 24/29, y = -12/29 और z = 18/29

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