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# Complex numbers

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25
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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
0

a/b = -2/7.

Jun 12, 2022
#2
0

No that gives 45+28i, not purely imaginary.

Guest Jun 12, 2022
#3
+1819
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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