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Let $a$ and $b$ be nonzero real numbers such that $$(2-7i)(a+bi)$$
is pure imaginary. Find $a/b.$

 Jun 12, 2022
 #1
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a/b = -2/7.

 Jun 12, 2022
 #2
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No that gives 45+28i, not purely imaginary. 

Guest Jun 12, 2022
 #3
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Evaluating, we get: \(2a - 2bi - 7ai +7b\) (recall that \(i^2 = -1\))

 

This means \(2a + 7b = 0\), that way, the real parts cancel out. 

 

Subtracting 2a from both sides, we get: \(7b = -2a\)

 

Solving for a gives us \(-{7 \over 2}b = a\), meaning \({a \over b} = \color{brown} \boxed {-{7 \over 2}}\)

 Jun 13, 2022

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