Solution 3.1:3

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The expression has real part equal to zero when <math>6+a=0</math>, i.e. <math>a=-6</math>.
The expression has real part equal to zero when <math>6+a=0</math>, i.e. <math>a=-6</math>.
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NOTE: Think about how to solve the problem is a is not a real number.
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NOTE: Think about how to solve the problem if a is not a real number.
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Revision as of 16:13, 25 September 2008

In order to be able to see the expression's real and imaginary parts directly, we treat it as an ordinary quotient of two complex numbers and multiply top and bottom by the complex conjugate of the numerator:


\displaystyle \begin{align}\frac{3+i}{2+ai}&=\frac{(3+i)(2-ai)}{(2+ai)(2-ai)}\\ &=\frac{3\cdot 2-3\cdot ai +i\cdot 2-ai^2}{2^2-(ai)^2}\\ &=\frac{6+a+(2-3a)i}{4+a^2}\\ &=\frac{6+a}{4+a^2}+\frac{2-3a}{4+a^2}i.\end{align}


The expression has real part equal to zero when \displaystyle 6+a=0, i.e. \displaystyle a=-6.

NOTE: Think about how to solve the problem if a is not a real number.

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