If $u+i v=\frac{3 i}{x+i y+2}$, then $y=$

If $u+i v=\frac{3 i}{x+i y+2}$, then $y=$
  1. $\frac{9 u}{u^2+v^2}$
  2. $\frac{3 u}{u^2+v^2}$
  3. $\frac{6 u}{u^2+v^2}$
  4. $\frac{12 u}{u^2+v^2}$

Solution

$ \begin{aligned} & \text {We have, } u+i v=\frac{3 i}{x+i y+2} \\ & \Rightarrow x+i y+2=\frac{3 i}{u+i v} \\ & \Rightarrow(x+2)+i y=\frac{3 i}{u+i v}+\frac{u-i v}{u-i v}=\frac{3 u i-3 v\left(i^2\right)}{u^2-(i v)^2} \\ & \Rightarrow(x+2)+i y=\frac{3 u i+3 v}{u^2+v^2} \\ & \Rightarrow(x+2)+i y=\frac{3 v}{u^2+v^2}+\frac{3 u}{u^2+v^2} i \end{aligned} $ by comparing Imaginary parts $ y=\frac{3 u}{u^2+v^2} $

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

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