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Last updated at Feb. 1, 2020 by Teachoo
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Example 29 Show that the lines (๐ฅ โ ๐ + ๐)/(๐ผ โ ๐ฟ) = (๐ฆ โ ๐)/๐ผ = (๐ง โ ๐ โ ๐)/(๐ผ + ๐ฟ) nd (๐ฅ โ ๐ + ๐)/(๐ฝ โ ๐พ) = (๐ฆ โ ๐)/๐ฝ = (๐ง โ ๐ โ ๐)/(๐ฝ + ๐พ) are coplanar.Two lines (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 and (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_2)/๐_2 are coplanar if |โ 8(๐_๐โ ๐_๐&๐_๐โ๐_๐&๐_๐โ๐_๐@๐_๐&๐_๐&๐_๐@๐_๐&๐_๐&๐_๐ )| = 0 (๐ โ ๐ + ๐ )/(๐ถ โ ๐น) = (๐ โ ๐)/๐ถ = (๐ โ ๐ โ ๐ )/(๐ถ + ๐น) (๐ฅ โ (๐ โ ๐))/(๐ผ โ ๐ฟ) = (๐ฆ โ ๐)/๐ผ = (๐ง โ (๐ + ๐))/(๐ผ + ๐ฟ) Comparing (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 ๐ฅ_1 = ๐ โ d , ๐ฆ_1= ๐ , ๐ง_1= ๐ + d & ๐_1=๐ผโ๐ฟ, ๐_1= ๐ผ, ๐_1= ๐ผ+๐ฟ (๐ โ ๐ + ๐)/(๐ท โ ๐ธ) = (๐ โ ๐)/๐ท = (๐ โ ๐ โ ๐)/(๐ท + ๐ธ) (๐ฅ โ (๐ โ ๐))/(๐ฝ โ ๐พ) = (๐ฆ โ ๐)/๐ฝ = (๐ง โ (๐ + ๐))/(๐ฝ + ๐พ) Comparing (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_1)/๐_2 ๐ฅ_2 = ๐ โ c , ๐ฆ_2= ๐ , ๐ง_2= ๐ + c & ๐_2 = ๐ฝโ๐พ, ๐_2 = ๐ฝ, ๐_2 = ๐ฝ + ๐พ Now, |โ 8(๐ฅ_2โ๐ฅ_1&๐ฆ_2โ๐ฆ_1&๐ง_2โ๐ง_1@๐_1&๐_1&๐_1@๐_2&๐_2&๐_2 )| = |โ 8(๐โ๐โ๐ + ๐&๐โ๐&๐+๐โ๐โ๐@๐ผโ๐ฟ&๐ผ&๐ผ+๐ฟ@๐ฝโ๐พ&๐ฝ&๐ฝ+๐พ)| Adding Column 3 to Column 1, = |โ 8(๐โ๐โ๐ + ๐+(๐+๐โ๐โ๐)&๐โ๐&๐+๐โ๐โ๐@๐ผโ๐ฟ+(๐ผ+๐ฟ)&๐ผ&๐ผ+๐ฟ@๐ฝโ๐พ+(๐ฝ+๐พ)&๐ฝ&๐ฝ+๐พ)| = |โ 8(2(๐โ๐)&๐โ๐&๐+๐โ๐โ๐@2๐ผ&๐ผ&๐ผ+๐ฟ@2๐ฝ&๐ฝ&๐ฝ+๐พ)| Taking 2 common from Column 1 = 2 |โ 8(๐ โ ๐&๐ โ ๐&๐ + ๐ โ ๐ โ ๐@๐ผ&๐ผ&๐ผ + ๐ฟ@๐ฝ&๐ฝ&๐ฝ +๐พ)| = 2 ร 0 = 0 Therefore, the given two lines are coplanar. Since Columns 1 and 2 are same, The value of determinant is zero.
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Example 10 Not in Syllabus - CBSE Exams 2021
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Example 23 Important Not in Syllabus - CBSE Exams 2021
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