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Misc 14 - Find x and y if (x - iy)(3 + 5i) is conjugate - Conjugate

Misc 14 - Chapter 5 Class 11 Complex Numbers - Part 2
Misc 14 - Chapter 5 Class 11 Complex Numbers - Part 3
Misc 14 - Chapter 5 Class 11 Complex Numbers - Part 4

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Misc 14 Find the real numbers x and y if (π‘₯ – 𝑖𝑦) (3 + 5𝑖) is the conjugate of –6 – 24𝑖. Conjugate of βˆ’6 βˆ’24𝑖 = βˆ’ 6 + 24𝑖 Now it is given that (π‘₯ – 𝑖𝑦) (3 + 5𝑖) is conjugate of βˆ’6 + 24𝑖 Hence from (1) and (2) βˆ’ 6 + 24𝑖 = (π‘₯ – 𝑖𝑦) (3 + 5𝑖) βˆ’ 6 + 24𝑖 = π‘₯ ( 3 + 5𝑖 ) – 𝑖𝑦 ( 3 + 5𝑖) βˆ’ 6 + 24𝑖 = 3π‘₯ + 5π‘₯𝑖 βˆ’ 3𝑦𝑖 – 5𝑖2𝑦 Putting 𝑖2 = –1 βˆ’ 6 + 24𝑖 = 3π‘₯ + 5π‘₯𝑖 βˆ’ 3𝑦𝑖 – 5 Γ— βˆ’ 1 Γ— 𝑦 βˆ’ 6 + 24𝑖 = 3π‘₯ + 5π‘₯𝑖 βˆ’ 3𝑦𝑖 + 5𝑦 βˆ’ 6 + 24𝑖 = 3π‘₯ + 5𝑦 βˆ’ 3𝑦𝑖 + 5π‘₯𝑖 βˆ’ 6 + 24𝑖 = 3π‘₯ + 5𝑦 + ( βˆ’ 3𝑦 + 5π‘₯)𝑖 Comparing real parts βˆ’ 6 = 3π‘₯ + 5𝑦 Comparing imaginary parts 24 = 5π‘₯ – 3𝑦 We solve equation (3) and (4) to find the value of x and y From (3) 3π‘₯ + 5𝑦 = βˆ’6 3π‘₯ = βˆ’6 βˆ’ 5𝑦 π‘₯ = (βˆ’6 βˆ’ 5𝑦)/3 Putting π‘₯ = (βˆ’6 βˆ’ 5𝑦)/3 in (4) 5π‘₯ – 3𝑦 = 24 5 ((βˆ’6 βˆ’ 5𝑦)/3) – 3𝑦 = 24 Multiplying 3 both sides 3Γ—5((βˆ’6 βˆ’ 5𝑦)/3) – 3Γ—3𝑦 = 3Γ—24 5(βˆ’6 βˆ’ 5𝑦) – 9𝑦 = 72 5 (βˆ’ 6) – 5 (5𝑦) – 9𝑦 = 72 βˆ’30 – 25π‘¦βˆ’ 9𝑦 = 72 βˆ’ 25𝑦 – 9𝑦 = 72 + 30 – 34𝑦 = 102 𝑦 = 102/(βˆ’34) 𝑦 = βˆ’3 Putting y = βˆ’ 3 in (4) 24 = 5π‘₯ – 3𝑦 24 = 5π‘₯ – 3(βˆ’3) 24 = 5π‘₯+9 24 βˆ’9= 5π‘₯ 15= 5π‘₯ 5π‘₯=15 π‘₯ = 15/5 π‘₯ = 3 Hence value of π‘₯ = 3 and 𝑦 = βˆ’3

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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.