Suppose `a^(2) = 5a - 8 and b^(2) =5b - 8` , then equation whose roots are `(a)/(b) and (b)/(a)` is

Options

A. `6x^(2) - 5x + 6 = 0`
B. `8x^(2) - 9x + 8 = 0`
C. `9x^(2) - 8x + 9 = 0`
D. `8x^(2) + 9x + 8 = 0`

Answer

To solve the problem, we start with the given equations:

1. \( a^2 = 5a - 8 \)
2. \( b^2 = 5b - 8 \)

We want to find the equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \).

### Step 1: Rearranging the equations
First, we can rearrange the given equations into standard quadratic form:

\[ a^2 - 5a + 8 = 0 \]
\[ b^2 - 5b + 8 = 0 \]

### Step 2: Finding the sum and product of the roots
For the quadratic equation \( x^2 - 5x + 8 = 0 \):
- The sum of the roots \( a + b \) can be calculated using the formula:
\[ \text{Sum of roots} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{-5}{1} = 5 \]

- The product of the roots \( ab \) can be calculated using the formula:
\[ \text{Product of roots} = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{8}{1} = 8 \]

### Step 3: Finding the sum and product of the new roots
Now, we need to find the sum and product of the new roots \( \frac{a}{b} \) and \( \frac{b}{a} \).

- The sum of the new roots is:
\[ \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \]

Using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \):
\[ a^2 + b^2 = (5)^2 - 2(8) = 25 - 16 = 9 \]
Thus,
\[ \frac{a}{b} + \frac{b}{a} = \frac{9}{8} \]

- The product of the new roots is:
\[ \frac{a}{b} \cdot \frac{b}{a} = 1 \]

### Step 4: Forming the quadratic equation
Now we can form the quadratic equation with the roots \( \frac{a}{b} \) and \( \frac{b}{a} \):
\[ x^2 - \left(\frac{a}{b} + \frac{b}{a}\right)x + \left(\frac{a}{b} \cdot \frac{b}{a}\right) = 0 \]

Substituting the values we found:
\[ x^2 - \left(\frac{9}{8}\right)x + 1 = 0 \]

### Step 5: Clearing the fraction
To eliminate the fraction, we can multiply the entire equation by 8:
\[ 8x^2 - 9x + 8 = 0 \]

### Final Answer
The equation whose roots are \( \frac{a}{b} \) and \( \frac{b}{a} \) is:
\[ 8x^2 - 9x + 8 = 0 \]