Question
Download Solution PDFनिम्नलिखित में से कौनसी स्थिति के तहत रेखाएं x = ay + b; z = cy + d और x = ey + f; z = gy + h लंबवत हैं?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFधारणा:
- यदि दो रेखाएं एक-दूसरे के लंबवत हैं तो इन रेखाओं के दिशा अनुपात का गुणनफल शून्य होता है।
गणना:
दी गई रेखाएं x = ay + b और z = cy + d हैं
⇒ x – b = ay और z – d = cy
\(\therefore {\rm{\;}}\frac{{{\rm{x}} - {\rm{b}}}}{{\rm{a}}} = {\rm{\;}}\frac{{\rm{y}}}{1} = {\rm{\;}}\frac{{{\rm{z}} - {\rm{d}}}}{{\rm{c}}}\) …. (1)
रेखा का दिशा अनुपात (a, 1, c) है
फिर से, रेखाएं x = ey + f और z = gy + h हैं
⇒ x – f = ey और z – h = gy
\(\therefore {\rm{\;}}\frac{{{\rm{x}} - {\rm{f}}}}{{\rm{e}}} = {\rm{\;}}\frac{{\rm{y}}}{1} = {\rm{\;}}\frac{{{\rm{z}} - {\rm{h}}}}{{\rm{g}}}\) …. (2)
रेखा का दिशा अनुपात (e, 1, g) है
दी गई दोनों रेखाएँ लंबवत हैं
∴ (a × e) + (1 × 1) + (c × g) = 0
⇒ ae + 1 + cg = 0
या ae + cg + 1 = 0
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