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

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

Transcript

Misc 14 Find the real numbers x and y if (๐‘ฅ โ€“ ๐‘–๐‘ฆ) (3 + 5๐‘–) is the conjugate of โ€“6 โ€“ 24๐‘–. Conjugate of โˆ’6 โˆ’24๐‘– = โˆ’ 6 + 24๐‘– Now it is given that (๐‘ฅ โ€“ ๐‘–๐‘ฆ) (3 + 5๐‘–) is conjugate of โˆ’6 + 24๐‘– Hence from (1) and (2) โˆ’ 6 + 24๐‘– = (๐‘ฅ โ€“ ๐‘–๐‘ฆ) (3 + 5๐‘–) โˆ’ 6 + 24๐‘– = ๐‘ฅ ( 3 + 5๐‘– ) โ€“ ๐‘–๐‘ฆ ( 3 + 5๐‘–) โˆ’ 6 + 24๐‘– = 3๐‘ฅ + 5๐‘ฅ๐‘– โˆ’ 3๐‘ฆ๐‘– โ€“ 5๐‘–2๐‘ฆ Putting ๐‘–2 = โ€“1 โˆ’ 6 + 24๐‘– = 3๐‘ฅ + 5๐‘ฅ๐‘– โˆ’ 3๐‘ฆ๐‘– โ€“ 5 ร— โˆ’ 1 ร— ๐‘ฆ โˆ’ 6 + 24๐‘– = 3๐‘ฅ + 5๐‘ฅ๐‘– โˆ’ 3๐‘ฆ๐‘– + 5๐‘ฆ โˆ’ 6 + 24๐‘– = 3๐‘ฅ + 5๐‘ฆ โˆ’ 3๐‘ฆ๐‘– + 5๐‘ฅ๐‘– โˆ’ 6 + 24๐‘– = 3๐‘ฅ + 5๐‘ฆ + ( โˆ’ 3๐‘ฆ + 5๐‘ฅ)๐‘– Comparing real parts โˆ’ 6 = 3๐‘ฅ + 5๐‘ฆ Comparing imaginary parts 24 = 5๐‘ฅ โ€“ 3๐‘ฆ We solve equation (3) and (4) to find the value of x and y From (3) 3๐‘ฅ + 5๐‘ฆ = โˆ’6 3๐‘ฅ = โˆ’6 โˆ’ 5๐‘ฆ ๐‘ฅ = (โˆ’6 โˆ’ 5๐‘ฆ)/3 Putting ๐‘ฅ = (โˆ’6 โˆ’ 5๐‘ฆ)/3 in (4) 5๐‘ฅ โ€“ 3๐‘ฆ = 24 5 ((โˆ’6 โˆ’ 5๐‘ฆ)/3) โ€“ 3๐‘ฆ = 24 Multiplying 3 both sides 3ร—5((โˆ’6 โˆ’ 5๐‘ฆ)/3) โ€“ 3ร—3๐‘ฆ = 3ร—24 5(โˆ’6 โˆ’ 5๐‘ฆ) โ€“ 9๐‘ฆ = 72 5 (โˆ’ 6) โ€“ 5 (5๐‘ฆ) โ€“ 9๐‘ฆ = 72 โˆ’30 โ€“ 25๐‘ฆโˆ’ 9๐‘ฆ = 72 โˆ’ 25๐‘ฆ โ€“ 9๐‘ฆ = 72 + 30 โ€“ 34๐‘ฆ = 102 ๐‘ฆ = 102/(โˆ’34) ๐‘ฆ = โˆ’3 Putting y = โˆ’ 3 in (4) 24 = 5๐‘ฅ โ€“ 3๐‘ฆ 24 = 5๐‘ฅ โ€“ 3(โˆ’3) 24 = 5๐‘ฅ+9 24 โˆ’9= 5๐‘ฅ 15= 5๐‘ฅ 5๐‘ฅ=15 ๐‘ฅ = 15/5 ๐‘ฅ = 3 Hence value of ๐‘ฅ = 3 and ๐‘ฆ = โˆ’3

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.