If `a^2=5a-3` and `b^2=5b-3,(a neb)` find the quadratic equation whose roots are `a/b` and `b/a`

Options

A. `3x^2-19x+1=0`
B. `2x^2=39x+27=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 \( \frac{b}{a} \), we start with the given equations for \( a \) and \( b \):

1. **Given Equations**:
\[ a^2 = 5a - 3 \]
\[ b^2 = 5b - 3 \]

2. **Rearranging the Equations**:
We can rearrange these equations into standard quadratic form:
\[ a^2 - 5a + 3 = 0 \]
\[ b^2 - 5b + 3 = 0 \]

3. **Identifying Roots**:
From the above equations, we see that \( a \) and \( b \) are the roots of the quadratic equation \( x^2 - 5x + 3 = 0 \).

4. **Finding the Sum and Product of Roots**:
Using Vieta's formulas, we can determine:
- Sum of roots \( a + b = 5 \)
- Product of roots \( ab = 3 \)

5. **Finding the Sum and Product of New Roots**:
We need to find the quadratic equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \).
- The sum of the new roots:
\[ \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \]
- The product of the new roots:
\[ \frac{a}{b} \cdot \frac{b}{a} = 1 \]

6. **Calculating \( a^2 + b^2 \)**:
We can express \( a^2 + b^2 \) using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting the values we found:
\[ a^2 + b^2 = 5^2 - 2 \cdot 3 = 25 - 6 = 19 \]

7. **Sum of New Roots**:
Now we can calculate the sum of the new roots:
\[ \frac{a^2 + b^2}{ab} = \frac{19}{3} \]

8. **Forming the Quadratic Equation**:
The quadratic equation with roots \( \frac{a}{b} \) and \( \frac{b}{a} \) can be formed using the sum and product of the roots:
\[ x^2 - \left(\frac{19}{3}\right)x + 1 = 0 \]

9. **Clearing the Fraction**:
To eliminate the fraction, we multiply the entire equation by 3:
\[ 3x^2 - 19x + 3 = 0 \]

10. **Final Result**:
The required quadratic equation is:
\[ 3x^2 - 19x + 3 = 0 \]