If `alpha,beta` are roots of equation `7x^2-3x+2=0` then findthe value of `alpha/(1-alpha^2)+beta/(1-beta^2)`

Options

A. `7/24`
B. `5/24`
C. `24/5`
D. `24/7`

Answer

To solve the problem, we need to find the value of \(\frac{\alpha}{1 - \alpha^2} + \frac{\beta}{1 - \beta^2}\), where \(\alpha\) and \(\beta\) are the roots of the equation \(7x^2 - 3x + 2 = 0\).

### Step 1: Find the roots \(\alpha\) and \(\beta\)
Using Vieta's formulas, we know:
- \(\alpha + \beta = -\frac{b}{a} = \frac{3}{7}\)
- \(\alpha \beta = \frac{c}{a} = \frac{2}{7}\)

### Step 2: Rewrite the expression
We want to find:
\[ \frac{\alpha}{1 - \alpha^2} + \frac{\beta}{1 - \beta^2} \]
We can rewrite this as:
\[ \frac{\alpha(1 - \beta^2) + \beta(1 - \alpha^2)}{(1 - \alpha^2)(1 - \beta^2)} \]
This simplifies to:
\[ \frac{\alpha + \beta - \alpha \beta^2 - \beta \alpha^2}{(1 - \alpha^2)(1 - \beta^2)} \]

### Step 3: Substitute known values
Substituting \(\alpha + \beta\) and \(\alpha \beta\):
\[ \frac{\frac{3}{7} - \alpha \beta(\alpha + \beta)}{(1 - \alpha^2)(1 - \beta^2)} \]
Since \(\alpha \beta = \frac{2}{7}\) and \(\alpha + \beta = \frac{3}{7}\), we have:
\[ \frac{\frac{3}{7} - \frac{2}{7} \cdot \frac{3}{7}}{(1 - \alpha^2)(1 - \beta^2)} \]
Calculating the numerator:
\[ \frac{3}{7} - \frac{6}{49} = \frac{21}{49} - \frac{6}{49} = \frac{15}{49} \]

### Step 4: Calculate the denominator
Now, we need to calculate \((1 - \alpha^2)(1 - \beta^2)\):
\[ (1 - \alpha^2)(1 - \beta^2) = 1 - (\alpha^2 + \beta^2) + \alpha^2 \beta^2 \]
We know:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta = \left(\frac{3}{7}\right)^2 - 2\left(\frac{2}{7}\right) = \frac{9}{49} - \frac{4}{7} = \frac{9}{49} - \frac{28}{49} = -\frac{19}{49} \]
And:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49} \]
Thus:
\[ (1 - \alpha^2)(1 - \beta^2) = 1 - \left(-\frac{19}{49}\right) + \frac{4}{49} = 1 + \frac{19}{49} + \frac{4}{49} = 1 + \frac{23}{49} = \frac{49}{49} + \frac{23}{49} = \frac{72}{49} \]

### Step 5: Combine results
Now we can combine the results:
\[ \frac{\frac{15}{49}}{\frac{72}{49}} = \frac{15}{72} = \frac{5}{24} \]

### Final Answer
Thus, the value of \(\frac{\alpha}{1 - \alpha^2} + \frac{\beta}{1 - \beta^2}\) is \(\frac{5}{24}\).

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