If 0 > a > b > c and the roots alpha, beta are imaginary roots of a x^2 + b x + c = 0 then |alpha| = |beta| |alpha| > 1 |beta| < 1 alpha + beta = 0

Given the quadratic equation $a x^{2} + b x + c = 0$ with $a < 0$, $b < 0$, and $c < 0$, and the roots $\alpha$ and $\beta$ are imaginary, we need to determine which of the following options is correct:

• $|\alpha| = |\beta|$
• $|\alpha| > 1$
• $|\beta| < 1$
• $\alpha + \beta = 0$

To solve this, we begin by noting that imaginary roots of a quadratic equation with real coefficients always occur in conjugate pairs. Hence, if $\alpha$ is $A + Bi$, then $\beta$ is its conjugate $A - Bi$.

Given: $$\alpha = A + Bi$$ $$\beta = A - Bi$$

We need to determine their magnitudes. The magnitude (absolute value) of a complex number $z = A + Bi$ is given by:

$$|z| = \sqrt{A^2 + B^2}$$

Thus:

$$|\alpha| = \sqrt{A^2 + B^2}$$ $$|\beta| = \sqrt{A^2 + B^2}$$

Since both magnitudes are calculated using the same formula, their magnitudes are equal, i.e., $|\alpha| = |\beta|$.

Thus, the correct option is:

Option A: $|\alpha| = |\beta|$

This concludes that both $|\alpha|$ and $|\beta|$ being equal satisfies the given conditions.