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Question
The complex numbers u v, , and w are related by `1/u = 1/v + 1/w`. If v = 3 – 4i and w = 4 + 3i, find u in rectangular form
Solution
v = 3 – 4i
w = 4 + 3i
Given Relation `1/u = 1/v + 1/w`
`1/u = 1/(3 - 4"i") + 1/(4 + 3"i")`
= `(1/(3 - 4"i") xx (3 + 4"i")/(3 + 4"i")) + (1/(4 + 3"i") xx (4 - 3"i")/(4 - 3"i"))`
= `(3 + 4"i")/((3)^2 - (4"i")^2) + (4 - 3"i")/((4)^2 - (3"i")^2`
= `(3 + 4"i")/(9 + 16) + (4 - 3"i")/(16 + 9)`
= `(3 + 4"i" + 4 - 3"i")/25`
= `(7 + "i")/25`
u = `25/(7 + "i") xx (7 - "i")/(7 - "i")`
= `(25(7 - "i"))/((7)^2 - "i"^2)`
= `(25(7 - "i"))/(49 + 1)`
= `25/50 (7 - "i")`
= `1/2 (7 - "i")`
= `7/2 - "i"/2`
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