Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers2 Marks
Question
Find the real values of x and y, if $(\text{x}+\text{iy})(2-3\text{i})=4+\text{i}$
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Answer
We have $(\text{x}+\text{iy})(2-3\text{i})=4+\text{i}$ $\Rightarrow\text{x}(2-3\text{i})+\text{iy}(2-3\text{i})=4+\text{i}$ $\Rightarrow2\text{x}-3\text{xi}+2\text{iy}+3\text{y}=4+\text{i}$ $\Rightarrow2\text{x}+3\text{y}+\text{i}(-3\text{x}+2\text{y})=4+\text{i}$ Equating the real and imaginary parts we get $2\text{x}+3\text{y}=4 \ ...(\text{i})$ $-3\text{x}+2\text{y}=1 \ ...(\text{ii})$ Multiplying (i) by 3 and (ii) by 2 and adding $6\text{x}-6\text{x}-9\text{y}+4\text{y}=12+2$ $\Rightarrow13\text{y}=14$ $\Rightarrow\text{y}=\frac{14}{13}$ Substituting the value of y in (i), we get $2\text{x}+3\times\frac{14}{13}=4$ $\Rightarrow2\text{x}+\frac{42}{13}=4$ $\Rightarrow2\text{x}=4-\frac{42}{13}$ $\Rightarrow2\text{x}=\frac{52-42}{13}$ $\Rightarrow2\text{x}=\frac{10}{13}$ $\Rightarrow\text{x}=\frac{5}{13}$ Hence $\text{x}=\frac{5}{13}$ and $\text{y}=\frac{14}{13}$
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