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Question
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values of a and b: `("i"(4 + 3"i"))/((1 - "i"))`
Solution
`("i"(4 + 3"i"))/(1 - "i") = (4"i" + 3"i"^2)/(1 - "i")`
= `(-3 + 4"i")/(1 - "i")` ...[∵ i2 = – 1]
= `((-3 + 4"i")(1 + "i"))/((1 - "i")(1 + "i")`
= `(-3 - 3"i" + 4"i" + 4"i"^2)/(1 - "i"^2)`
= `(-3 + "i" + 4(-1))/(1 - (-1)` ...[∵ i2 = – 1]
= `(-7 + "i")/2`
∴ `("i"(4 + 3"i"))/(1 - "i") = (-7)/2 + 1/2 "i"`
∴ a = `(-7)/2 and "b" = 1/2`.
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