Question

If $0>a>b>c$ and the roots $\alpha, \beta$ are imaginary roots of $a x^{2}+b x+c=0$ then

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

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Answer

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.


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