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Complex Numbers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSComplex Numbers2 Marks
Question
Express the following complex numbers in the standard form a + ib: $\frac{1}{(2+\text{i})^2}$
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Answer

$\frac{1}{(2+\text{i})^2}=\frac{1}{2^2+(\text{i})^2+2\times2\times\text{i}}$ $=\frac{1}{4-1+4\text{i}}$ $=\frac{1}{3+4\text{i}}$ $\frac{1}{(3+4\text{i})}\times\frac{(3-4\text{i})}{(3-4\text{i})}$ [On rationalising the denominator] $=\frac{3-4\text{i}}{3^2+4^2} \ \big[\because \ (\text{a}+\text{ib})(\text{a}-\text{ib})=\text{a}^2+\text{b}^2\big]$ $=\frac{3-4\text{i}}{25}$ $=\frac{3}{25}-\frac{4}{25}\text{i}$ $\therefore \ \frac{1}{(2+\text{i})^2}=\frac{3}{25}-\frac{4}{25}\text{i}$

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