Let $a$ and $b$ be nonzero real numbers such that $$(2-7i)(a+bi)$$
is pure imaginary. Find $a/b.$
+2666 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 \({a \over b} = \color{brown} \boxed {-{7 \over 2}}\)
