If $5 z_{2} / 11 z_{1}$ is purely imaginary, then the value of $| (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) |$ is

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Solution

Finding the value of $| (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) |$ :
Given that $5 z_{2} / 11 z_{1}$ is purely imaginary
Let us take $5 z_{2} / 11 z_{1} = i y$
$z_{2} / z_{1} = 11 i y / 5 | (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) = z_{1} (2 + 3 z_{2} / z_{1}) / z_{1} (2 - 3 z_{2} / z_{1}) = (2 + 3 (11 i y / 5)) / (2 - 3 (11 i y / 3)) (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) = (10 + 33 i y) / (10 - 33 i y) | (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) | = \sqrt{(100 + 1089)} / \sqrt{(100 + 1089)} = 1$
Hence, the value of $| (2 z_{1} + 3 z_{2}) / (2 z_{1} - 3 z_{2}) |$ is $1$ .

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