If α≠β and \(\alpha^2=5\alpha-3,\beta^2=5\beta-3\), then the equation whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) is
Step-by-step Solution:
The equation that is satisfied by both \( \alpha \) and \( \beta \) is: \[ x^2 = 5x - 3 \quad \text{or} \quad x^2 - 5x + 3 = 0 \] Now, the sum of the roots is: \[ \frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta} = \frac{(\alpha + \beta)^2 - 2\alpha \beta}{\alpha \beta} \] \[ = \frac{25 - 2 \times 3}{3} = \frac{19}{3} \] The product of the roots is 1, so the equation is: \[ x^2 - \frac{19}{3}x + 1 = 0 \quad \text{or} \quad 3x^2 - 19x + 3 = 0 \]