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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 8 If x + iy = (a + ib)﷮(𝑎 − 𝑖𝑏)﷯ Taking R.H.S a + ib﷮𝑎 − 𝑖𝑏﷯ Rationalizing = a + ib﷮𝑎 − 𝑖𝑏﷯ (a + ib)﷮(𝑎+ 𝑖𝑏)﷯ = (a + ib)﷮2﷯﷮ 𝑎 − 𝑖𝑏﷯ (𝑎 + 𝑖𝑏)﷯ = 𝑎﷮2﷯ + (𝑖𝑏)﷮2﷯ + 2𝑎𝑖𝑏﷮ 𝑎﷮2﷯ − (𝑖𝑏)﷮2﷯﷯ = 𝑎﷮2﷯ + 𝑖﷮2﷯ 𝑏﷮2﷯ + 2𝑎𝑖𝑏﷮ 𝑎﷮2﷯ − 𝑖﷮2﷯﷯ ( 𝑏﷮2﷯)﷯ = 𝑎﷮2﷯ − 𝑏﷮2﷯ + 2𝑎𝑖𝑏﷮ 𝑎﷮2﷯ + 𝑏﷮2﷯﷯ = 𝑎﷮2﷯ − 𝑏﷮2﷯﷮ 𝑎﷮2﷯ + 𝑏﷮2﷯﷯ + i 2𝑎𝑏﷮ 𝑎﷮2﷯ + 𝑏﷮2﷯﷯﷯ Now L.H.S = x + iy Comparing real and imaginary parts in L.H.S and R.H.S Adding (1) and (2) 𝑥﷮2﷯+ 𝑦﷮2﷯= 𝑎﷮2﷯ − 𝑏﷮2﷯﷮ 𝑎﷮2﷯ + 𝑏﷮2﷯﷯﷯﷮2﷯+ (2𝑎𝑏)﷮2﷯﷮ ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷯ = 1﷮ ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷯ 𝑎﷮2﷯ − 𝑏﷮2﷯﷯﷮2﷯ + 2𝑎𝑏﷯﷮2﷯﷯ = 1﷮ ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷯ 𝑎﷮4﷯ + 𝑏﷮4﷯−2 𝑎﷮2﷯ 𝑏﷮2﷯+ 4𝑎﷮2﷯ 𝑏﷮2﷯﷯ = 𝑎﷮2﷯ ﷯﷮2﷯ + 𝑏﷮2﷯﷯﷮2﷯ + 2 𝑎﷮2﷯ 𝑏﷮2﷯﷮ ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷯ = ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷮ ( 𝑎﷮2﷯ + 𝑏﷮2﷯)﷮𝟐﷯﷯ = 1 Thus, 𝑥﷮2﷯+ 𝑦﷮2﷯=1 Hence, Proved

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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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.