If ` alpha ne beta ` but ` alpha^(2)= 5 alpha - 3 ` and ` beta ^(2)= 5 beta -3 ` then the equation having ` alpha // beta and beta // alpha ` as its roots is :

Options

A. ` 3 x^(2) - 19 x +3=0`
B. `3x^(2) + 19x -3 =0`
C. `3x^(2) - 19x -3 =0`
D. `x^(2) - 5x +3 =0 `

Answer

To solve the problem, we need to find the equation that has roots \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \). We start with the equations given for \( \alpha \) and \( \beta \):

1. \( \alpha^2 = 5\alpha - 3 \)
2. \( \beta^2 = 5\beta - 3 \)

### Step 1: Rearranging the equations
We can rearrange both equations:
- From the first equation: \( \alpha^2 - 5\alpha + 3 = 0 \)
- From the second equation: \( \beta^2 - 5\beta + 3 = 0 \)

### Step 2: Subtract the two equations
Subtract the first equation from the second:
\[ \beta^2 - \alpha^2 = 5\beta - 5\alpha \]
Factoring both sides gives:
\[ (\beta - \alpha)(\beta + \alpha) = 5(\beta - \alpha) \]
Since \( \alpha \neq \beta \), we can divide both sides by \( \beta - \alpha \):
\[ \beta + \alpha = 5 \]

### Step 3: Add the two equations
Now, add the two equations:
\[ \alpha^2 + \beta^2 = 5\alpha + 5\beta - 6 \]
Using the fact that \( \alpha + \beta = 5 \):
\[ \alpha^2 + \beta^2 = 5(5) - 6 = 25 - 6 = 19 \]

### Step 4: Finding \( \alpha \beta \)
We know:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the known values:
\[ 19 = 25 - 2\alpha\beta \]
Rearranging gives:
\[ 2\alpha\beta = 25 - 19 = 6 \quad \Rightarrow \quad \alpha\beta = 3 \]

### Step 5: Forming the required equation
The roots of the equation are \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \). The sum of the roots is:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{19}{3} \]
The product of the roots is:
\[ \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 6: Writing the quadratic equation
Using the standard form \( x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \):
\[ x^2 - \frac{19}{3}x + 1 = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ 3x^2 - 19x + 3 = 0 \]

### Final Answer
Thus, the required equation is:
\[ \boxed{3x^2 - 19x + 3 = 0} \]