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

  1. Chapter 5 Class 11 Complex Numbers
  2. Serial order wise
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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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