Check sibling questions

Example 14 - If x + iy = (a +  ib) / (a - ib), prove x2 + y2 = 1 - Examples

Example 14 - Chapter 5 Class 11 Complex Numbers - Part 2
Example 14 - Chapter 5 Class 11 Complex Numbers - Part 3


Transcript

Example 14 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.