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 $\alpha^{2}+\beta^{2}=$ ________.

• A. $3\alpha \beta$
• B. $\frac{1}{2}\alpha \beta$
• C. $\alpha \beta$
• D. $4\alpha \beta$
• E. $2\alpha \beta$

Hint:

$A.M. = G.M. \iff \alpha = \beta$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
The problem states that the Arithmetic Mean (A.M.) and the Geometric Mean (G.M.) of two numbers, \(\alpha\) and \(\beta\), are equal. This has a very specific implication for the relationship between \(\alpha\) and \(\beta\), which we can use to evaluate the expression \(\alpha^2 + \beta^2\).
Step 2: Key Formula or Approach
For two positive numbers \(\alpha\) and \(\beta\):
- Arithmetic Mean (A.M.) = \(\frac{\alpha + \beta}{2}\)
- Geometric Mean (G.M.) = \(\sqrt{\alpha\beta}\)
The AM-GM inequality states that A.M. \(\geq\) G.M., with equality holding if and only if the numbers are equal (\(\alpha = \beta\)).
Step 3: Detailed Explanation
1. Set up the given condition.
We are told that \(\lambda\) is both the A.M. and the G.M. of \(\alpha\) and \(\beta\).
\[ \text{A.M.} = \frac{\alpha + \beta}{2} \quad \text{and} \quad \text{G.M.} = \sqrt{\alpha\beta} \] Therefore, we have:
\[ \frac{\alpha + \beta}{2} = \sqrt{\alpha\beta} \] 2. Deduce the relationship between \(\alpha\) and \(\beta\).
The condition that the A.M. is equal to the G.M. is the case where equality holds in the AM-GM inequality. This happens only when the numbers themselves are equal.
\[ \alpha = \beta \] 3. Evaluate the expression \(\alpha^2 + \beta^2\).
Now we need to compute \(\alpha^2 + \beta^2\) using the fact that \(\alpha = \beta\).
Substitute \(\beta\) with \(\alpha\) in the expression:
\[ \alpha^2 + \beta^2 = \alpha^2 + (\alpha)^2 = 2\alpha^2 \] 4. Express the result in the format of the options.
The options are given in terms of \(\alpha\beta\). Since \(\alpha = \beta\), we can write our result, \(2\alpha^2\), as:
\[ 2\alpha^2 = 2 \cdot \alpha \cdot \alpha = 2 \cdot \alpha \cdot \beta \] So, \(\alpha^2 + \beta^2 = 2\alpha\beta\).
Step 4: Final Answer
The value of \(\alpha^2 + \beta^2\) is \(2\alpha\beta\).