Quadratic Equations Question - None

Question

If $\alpha \neq \beta$ but $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ then the equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

Accepted Answer

Step 1: Since $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$, both $\alpha$ and $\beta$ are roots of $x^2 = 5x - 3$ or $x^2 - 5x + 3 = 0$.

Step 2: By Vieta's formulas: $\alpha + \beta = 5$ and $\alpha\beta = 3$.

Step 3: For the new equation with roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$:

Sum of roots = $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha\beta} = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta} = \frac{25 - 6}{3} = \frac{19}{3}$

Product of roots = $\frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1$

Step 4: The equation is $x^2 - \frac{19}{3}x + 1 = 0$ or $3x^2 - 19x + 3 = 0$.

∴ Answer is (A).

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