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Misc 16 - If (x + iy)3 = u + iv, then show that u/x + v/y - Proof- Solving

Misc 16 - Chapter 5 Class 11 Complex Numbers - Part 2
Misc 16 - Chapter 5 Class 11 Complex Numbers - Part 3

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Misc, 16 If (x + iy)3 = u + iv, then show that u/x + v/y = 4 (𝑥2 – 𝑦2) . We know that (𝑎 + 𝑏)^3 = 𝑎3 + 𝑏3 +3𝑎𝑏 (𝑎 + 𝑏) Replacing a = x and b = iy (𝑥 + 𝑖𝑦)3= 𝑥3 + (𝑖𝑦)3 + 3 𝑥 𝑖𝑦 (𝑥 + 𝑖𝑦) = 𝑥3 + 𝑖3𝑦3 + 3𝑥 𝑦𝑖 (𝑥 + 𝑖𝑦) = 𝑥3 + 𝑖2 ×𝑖 𝑦3 + 3𝑥2𝑦𝑖+ 3𝑥𝑦2𝑖2 Putting 𝑖2 = –1 = 𝑥3 + (− 1 × 𝑖 × 𝑥𝑦2) + 3𝑥2 𝑦𝑖 + 3𝑥𝑦2 𝑥(−1) = 𝑥3 – 𝑖𝑦3 + 3𝑥2 𝑦𝑖 − 3𝑥𝑦2 = 𝑥3 – 3𝑥𝑦2 − 𝑖𝑦3 + 3𝑥2𝑦𝑖 = 𝑥3 – 3𝑥𝑦2 + 3𝑥2𝑦𝑖 − 𝑖𝑦3 = 𝑥3 – 3𝑥𝑦2 + (3𝑥2𝑦 − 𝑦3)𝑖 Hence, (𝑥 + 𝑖𝑦)3 = 𝑥3 – 3𝑥𝑦2 + (3𝑥2𝑦 − 𝑦3)𝑖 But, (𝑥 + 𝑖𝑦)3 = 𝑢 + 𝑖𝑣 So, 𝑥3 – 3𝑥𝑦2 + (3𝑥2𝑦 − 𝑦3)𝑖 = 𝑢 + 𝑖𝑣 Comparing Real parts 𝑥3 – 3𝑥𝑦2 = 𝑢 𝑥 (𝑥2– 3𝑦2) = 𝑢 𝑥2 – 3𝑦2 = 𝑢/𝑥 Adding (1) & (2) i.e. (1) + (2) 𝑢/𝑥 + 𝑣/𝑦 = (𝑥2 – 3𝑦2) + (3𝑥2 –𝑦2) = 𝑥2 – 3𝑦2 +3𝑥2 – 𝑦2 = 4𝑥2 – 4𝑦2 = 4 (𝑥2 – 𝑦2) Thus, u/x + v/y = 4 (x2 – y2) 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.