The quadratic equation whose roots are a/b and b/a, `a!=b!=0` where `a^(2)=5a-3`, and `b^(2)=5b-3`, is

Options

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

Answer

To find the quadratic equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \), given the conditions \( a^2 = 5a - 3 \) and \( b^2 = 5b - 3 \), we can follow these steps:

### Step 1: Find the values of \( a \) and \( b \)
From the equations \( a^2 = 5a - 3 \) and \( b^2 = 5b - 3 \), we can rewrite them as:
\[ a^2 - 5a + 3 = 0 \]
\[ b^2 - 5b + 3 = 0 \]

### Step 2: Calculate the roots of the quadratic equations
To find the roots of the quadratic equation \( x^2 - 5x + 3 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -5, c = 3 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 3}}{2 \cdot 1} = \frac{5 \pm \sqrt{25 - 12}}{2} = \frac{5 \pm \sqrt{13}}{2} \]
Thus, the roots are:
\[ a = \frac{5 + \sqrt{13}}{2}, \quad b = \frac{5 - \sqrt{13}}{2} \]

### Step 3: Find \( \alpha + \beta \) and \( \alpha \beta \)
The roots we need to find the quadratic for are \( \alpha = \frac{a}{b} \) and \( \beta = \frac{b}{a} \).

**Sum of the roots \( \alpha + \beta \)**:
\[ \alpha + \beta = \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \]

**Product of the roots \( \alpha \beta \)**:
\[ \alpha \beta = \frac{a}{b} \cdot \frac{b}{a} = 1 \]

### Step 4: Calculate \( a^2 + b^2 \) and \( ab \)
Using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):
- From \( a + b = 5 \) (sum of roots of the original quadratic)
- From \( ab = 3 \) (product of roots of the original quadratic)

Calculating \( a^2 + b^2 \):
\[ a^2 + b^2 = (5)^2 - 2 \cdot 3 = 25 - 6 = 19 \]

Now substituting into \( \alpha + \beta \):
\[ \alpha + \beta = \frac{19}{3} \]

### Step 5: Form the quadratic equation
Using the sum and product of the roots, the quadratic equation can be formed as:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values:
\[ x^2 - \frac{19}{3}x + 1 = 0 \]

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

### Final Answer
The quadratic equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \) is:
\[ \boxed{3x^2 - 19x + 3 = 0} \]