Question
Download Solution PDFK का मान जिसके लिए सरल रेखा x + y + 3z - 2 = 0 = 2x + y - z - 3 समतल 3x + 2y + kz - 4 = 0 के समानांतर है:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिया गया है:
सरल रेखा x + y + 3z - 2 = 0 = 2x + y - z - 3 समतल 3x + 2y + kz - 4 = 0 के समानांतर है।
संकल्पना:
यदि कोई रेखा समतल के समानांतर है तो रेखा का दिक्-अनुपात और समतल के अभिलंब का बिंदु गुणनफल 0 होना चाहिए।
गणना:
सरल रेखा x + y + 3z - 2 = 0 = 2x + y - z - 3 समतल 3x + 2y + kz - 4 = 0 के समानांतर है।
तो, समतलों के अभिलंबों का दिक्-अनुपात क्रमशः (1, 1, 3) और (2, 1, -1) है।
इसलिए आवश्यक रेखा का दिक्-अनुपात इनके अभिलंब का बिंदु गुणनफल है
अर्थात \(\rm \begin{vmatrix}i&j&k\\1&1&3\\2&1&-1\end{vmatrix}\)
= i(-1 - 3) - j(-1 - 6) + k(1 - 2)
= - 4i + 7j - k
इसलिए, रेखा का दिक्-अनुपात (-4, 7, -1) है।
अब रेखा 3x + 2y + kz - 4 = 0 के समानांतर है, इसलिए रेखा का दिक्-अनुपात और समतल के अभिलंब का बिंदु गुणनफल 0 होना चाहिए।
∴ 3(-4) + 2(7) + k(-1) = 0
⇒ 2 - k = 0
⇒ k = 2
अतः विकल्प (2) सही है।
Last updated on Jul 16, 2025
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