Check sibling questions

Example 21 - Show that lines x+3/3 = y-1/1 = z-5/5 are coplanar

Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Transcript

Example 21 Show that the lines (π‘₯ + 3)/( βˆ’3) = (𝑦 βˆ’ 1)/1 = (𝑧 βˆ’ 5)/5 and (π‘₯ + 1)/( βˆ’1) = (𝑦 βˆ’ 2)/2 = (𝑧 βˆ’ 5)/5 are coplanar. Two lines (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 and (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_2)/𝑐_2 are coplanar if |β– 8(𝒙_πŸβˆ’π’™_𝟏&π’š_πŸβˆ’π’š_𝟏&𝒛_πŸβˆ’π’›_𝟏@𝒂_𝟏&𝒃_𝟏&𝒄_𝟏@𝒂_𝟐&𝒃_𝟐&𝒄_𝟐 )| = 0 Given, the two lines are Given, (𝒙 + πŸ‘)/( βˆ’ πŸ‘) = (π’š βˆ’ 𝟏)/𝟏 = (𝒛 βˆ’ πŸ“)/πŸ“ (π‘₯ βˆ’(βˆ’3))/( βˆ’ 3) = (𝑦 βˆ’1)/1 = (π‘§βˆ’ 5)/5 Comparing (π‘₯ βˆ’ π‘₯_1)/π‘Ž_1 = (𝑦 βˆ’ 𝑦_1)/𝑏_1 = (𝑧 βˆ’ 𝑧_1)/𝑐_1 π‘₯_1 = βˆ’3, 𝑦_1 = 1, 𝑧_1= 5 & π‘Ž_1 = βˆ’3, 𝑏_1 = 1, 𝑐_1= 5 Given, (𝒙 + 𝟏)/( βˆ’ 𝟏) = (π’š βˆ’ 𝟐)/𝟏 = (𝒛 βˆ’ πŸ“)/πŸ“ (π‘₯ βˆ’ (βˆ’1))/( βˆ’ 1) = (𝑦 βˆ’ 2)/2 = (𝑧 βˆ’ 5)/5 Comparing (π‘₯ βˆ’ π‘₯_2)/π‘Ž_2 = (𝑦 βˆ’ 𝑦_2)/𝑏_2 = (𝑧 βˆ’ 𝑧_2)/𝑐_2 π‘₯_2 = βˆ’1, 𝑦_2 = 2, 𝑧_2= 5 & π‘Ž_2 = βˆ’1, 𝑏_2 = 2, 𝑐_2= 5 Now, |β– 8(π‘₯_2βˆ’π‘₯_1&𝑦_2βˆ’π‘¦_1&𝑧_2βˆ’π‘§_1@π‘Ž_1&𝑏_1&𝑐_1@π‘Ž_2&𝑏_2&𝑐_2 )| `= |β– 8( βˆ’1βˆ’(βˆ’3)&2βˆ’1&5βˆ’5@ βˆ’3&1&5@ βˆ’1&2&5)| = |β– 8(2&1&0@βˆ’3&1&5@βˆ’1&2&5)| = 2[(1Γ—5)βˆ’(2Γ—5)] βˆ’ 1[(βˆ’3Γ—5)βˆ’(βˆ’ 1Γ—5)] + 0 [(βˆ’3Γ—2)βˆ’(βˆ’1Γ—1)] = 2[5βˆ’10]βˆ’ 1[βˆ’ 15βˆ’(βˆ’ 5)] + 0 = 2(βˆ’5) βˆ’1(βˆ’10) = βˆ’10 + 10 = 0 Therefore, the given two lines are coplanar.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.