Discussion
Which of the following complex numbers are collinear ?
Answer: A
The collinear points are \(1 + 2i, 2 + 5i, 4 + 11i\).
Three points \(z_{1}, z_{2}, z_{3}\) are collinear when \(z_{2} - z_{1}\) and \(z_{3} - z_{1}\) are real multiples of each other. Here
\[ \begin{aligned} z_{2} - z_{1} &= 1 + 3i \\ z_{3} - z_{1} &= 3 + 9i = 3(1 + 3i) \end{aligned} \]
so the points lie on the line \(y = 3x - 1\).
The other sets fail the test. For \(1 - i, 2 + i, 1 + i\) the differences are \(1 + 2i\) and \(2i\); for \(i, -1 + 2i, 3 + 4i\) they are \(-1 + i\) and \(3 + 3i\); for \(0, -3 + i, 7 + 8i\) they are \(-3 + i\) and \(7 + 8i\). In each case the ratio is not real.
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