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 step by step, we will derive the quadratic equation that has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

### Step 1: Understand the equations for \(\alpha\) and \(\beta\)
We are given:
\[ \alpha^2 = 5\alpha - 3 \]
\[ \beta^2 = 5\beta - 3 \]
Rearranging both equations gives us:
\[ \alpha^2 - 5\alpha + 3 = 0 \]
\[ \beta^2 - 5\beta + 3 = 0 \]

### Step 2: Form the quadratic equation with roots \(\alpha\) and \(\beta\)
From the standard form of a quadratic equation \(x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0\), we can find the sum and product of the roots:
- Sum of roots \((\alpha + \beta) = 5\)
- Product of roots \((\alpha \beta) = 3\)

Thus, the quadratic equation with roots \(\alpha\) and \(\beta\) is:
\[ x^2 - 5x + 3 = 0 \]

### Step 3: Find the new roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\)
Now we need to find the sum and product of the new roots:
- Sum of roots:
\[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} \]
Using the identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\):
\[ \alpha^2 + \beta^2 = 5^2 - 2 \cdot 3 = 25 - 6 = 19 \]
Thus,
\[ \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{19}{3} \]

- Product of roots:
\[ \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1 \]

### Step 4: Form the quadratic equation with new roots
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]
Substituting the values:
\[ x^2 - \frac{19}{3}x + 1 = 0 \]

To eliminate the fraction, multiply the entire equation by 3:
\[ 3x^2 - 19x + 3 = 0 \]

### Final Answer
The equation having \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) as its roots is:
\[ 3x^2 - 19x + 3 = 0 \]