Question

Find $\mathrm{a}$ and $\mathrm{b}$ if $(\mathrm{a}+\mathrm{ib})(1+\mathrm{i})=2+\mathrm{i}$.

Answer 1

Step 1: Simplify the given expression

Given, $(\mathrm{a}+\mathrm{ib})(1+\mathrm{i})=2+\mathrm{i}$

$$\begin{gathered} \Rightarrow a+ai+ib+i^{2}b=2+i \\ \Rightarrow a+ai+ib-b=2+i \\ \Rightarrow a-b+\left(a+b\right)i=2+i \end{gathered}$$

Step 2: Compare RHS and LHS and solve

$$\begin{gathered} \Rightarrow a-b=2\dots i \\ \Rightarrow a+b=1\dots ii \end{gathered}$$

Now, solving the two equations we get,

$$\begin{gathered} \Rightarrow 2a=3 \\ \Rightarrow a=\frac{3}{2} \end{gathered}$$

and,

$$\begin{gathered} \Rightarrow b=1-\frac{3}{2} \\ \Rightarrow b=-\frac{1}{2} \end{gathered}$$

Hence, the required answer is $a=\frac{3}{2},b=\frac{-1}{2}$.