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Problem 672

AMC 10/12, early questions

Algebra Difficulty 3.9 Find the answer China Mathematical Competition · China

Suppose $m$ is a real number, and complex numbers $z_1 = 1 + 2i$ , $z_2 = m + 3i$ , where $i$ is the imaginary unit. If $z_1 \cdot \bar{z_2}$ is purely imaginary, then the value of $|z_1 + z_2|$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Since $z_1 \cdot \bar{z_2} = (1 + 2i)(m - 3i) = m + 6 + (2m - 3)i$ is purely imaginary, we get $m = -6$ . Therefore, $|z_1 + z_2| = |-5 + 5i| = 5\sqrt{2}$ .

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