Question:
Let $\alpha$ and $\beta$ be the roots. If $A$ is the A.M. between $\alpha$ and $\beta$ and also $G$ is the G.M. between $\alpha$ and $\beta$, then $\alpha^{2}+\beta^{2}=$ ________.
Let $\alpha$ and $\beta$ be the roots. If $A$ is the A.M. between $\alpha$ and $\beta$ and also $G$ is the G.M. between $\alpha$ and $\beta$, then $\alpha^{2}+\beta^{2}=$ ________.
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$A.M. = G.M. \iff \alpha = \beta$.
Updated On: Jun 26, 2026
- $3\alpha \beta$
- $\frac{1}{2}\alpha \beta$
- $\alpha \beta$
- $4\alpha \beta$
- $2\alpha \beta$
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The Correct Option is
Solution and Explanation
Step 1: Concept
Note: The question paper source is fragmented. It asks for $\alpha^2 + \beta^2$ when $A.M. = G.M.$
Step 2: Meaning
If $A.M. = G.M.$ for two numbers, then the numbers must be equal ($\alpha = \beta$).
Step 3: Analysis
If $\alpha = \beta$, then $\alpha^2 + \beta^2 = \alpha^2 + \alpha^2 = 2\alpha^2$. Since $\alpha = \beta$, we can write $\alpha^2$ as $\alpha \cdot \beta$.
Step 4: Conclusion
$\alpha^2 + \beta^2 = 2\alpha\beta$. Final Answer: (E)
Note: The question paper source is fragmented. It asks for $\alpha^2 + \beta^2$ when $A.M. = G.M.$
Step 2: Meaning
If $A.M. = G.M.$ for two numbers, then the numbers must be equal ($\alpha = \beta$).
Step 3: Analysis
If $\alpha = \beta$, then $\alpha^2 + \beta^2 = \alpha^2 + \alpha^2 = 2\alpha^2$. Since $\alpha = \beta$, we can write $\alpha^2$ as $\alpha \cdot \beta$.
Step 4: Conclusion
$\alpha^2 + \beta^2 = 2\alpha\beta$. Final Answer: (E)
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