For the following two (02) items : Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 + (\log_{0.5} (\alpha^2))x + (\log_{0.5} (\alpha^2))^4 = 0$ where $\alpha^2 \neq 1$ and $\log_{0.5} (\alpha^2) > 0$. Further, $\beta^2 = \alpha (\log_{\alpha^2} (0.5))$
\(\alpha = -2\beta\)
To find the relation between \( \alpha \) and \( \beta \), given the quadratic equation and conditions, we start by considering the quadratic equation:
\(x^2 + (\log_{0.5} (\alpha^2))x + (\log_{0.5} (\alpha^2))^4 = 0\).
Since \( \alpha \) and \( \beta \) are roots of this quadratic equation, according to Vieta's formulas, the sum and product of the roots \( \alpha \) and \( \beta \) are given by:
\(\alpha + \beta = -\log_{0.5} (\alpha^2)\), and
\(\alpha \beta = (\log_{0.5} (\alpha^2))^4\).
Additionally, we are given that:
\(\beta^2 = \alpha \cdot \log_{\alpha^2} (0.5)\).
First, consider the simplification of \( \log_{\alpha^2} (0.5) \):
Using the change of base formula: \[ \log_{\alpha^2} (0.5) = \frac{\log_{0.5} (0.5)}{\log_{0.5} (\alpha^2)} \] Since \(\log_{0.5} (0.5) = 1\), it simplifies to: \[ \log_{\alpha^2} (0.5) = \frac{1}{\log_{0.5} (\alpha^2)} \]
Substituting this in the given condition:
\(\beta^2 = \alpha \cdot \frac{1}{\log_{0.5} (\alpha^2)}\).
From the product of roots \(\alpha \beta = (\log_{0.5} (\alpha^2))^4\), we have:
\(\alpha = \frac{(\log_{0.5} (\alpha^2))^4}{\beta}\).
Substitute \(\alpha\) into the equation for \(\beta^2\):
\[ \beta^2 = \frac{(\log_{0.5} (\alpha^2))^4}{\beta \cdot \log_{0.5} (\alpha^2)} \]
This simplifies to: \[ \beta^3 = \frac{(\log_{0.5} (\alpha^2))^3}{\log_{0.5} (\alpha^2)} \]
\(\beta^3 = (\log_{0.5} (\alpha^2))^3\).
Taking cube roots on both sides, since \(\log_{0.5} (\alpha^2) > 0\), we get:
\(\beta = \log_{0.5} (\alpha^2)\).
Substituting \(\beta = \log_{0.5} (\alpha^2)\) in the equation for the sum of roots:
\(\alpha + \beta = -\log_{0.5} (\alpha^2)\),
gives:
\(\alpha + \log_{0.5} (\alpha^2) = -\log_{0.5} (\alpha^2)\).
Solving for \(\alpha\), we find:
\(\alpha = -2\log_{0.5} (\alpha^2) = -2\beta\).
Thus, the relation between \( \alpha \) and \( \beta \) is:
\(\alpha = -2\beta\).
This matches the correct answer from the options provided.
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