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The circle in the complex plane given by z*conj(z) + conj(alpha)*z + alpha*conj(z) + r = 0 (with alpha complex and r real) cuts the real axis. Find the length of the intercept it makes on the real axis.

1. sqrt((alpha + conj(alpha)) - r)
2. sqrt((alpha + conj(alpha))² - 2r)
3. sqrt((alpha + conj(alpha))² + r)
4. sqrt((alpha + conj(alpha))² - 4r)

Correct answer: sqrt((alpha + conj(alpha))² - 4r)

Solution

Substituting z = x reduces the circle equation to x² + (alpha + conj(alpha))x + r = 0; the chord length on the real axis is the distance between its two real roots.

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