Question

If $\alpha$ and $\beta$ are non-zero distinct complex numbers satisfying $\alpha^2+3=5 \alpha$ and $\beta^2=5 \beta-3$ then a quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

• A. $3 x ^2-37 x +3=0$
• B. $3 x^2-31 x+3=0$
• C. $3 x^2-19 x+3=0$
• D. $3 x^2-17 x+3=0$

Answer 1

Answer: (C)

Given, $ \alpha^2=5 \alpha-3 \Rightarrow \alpha^2-5 \alpha+3=0 $
$\text { and } \beta^2=5 \beta-3 \Rightarrow \beta^2-5 \beta+3=0$

Now, $\alpha+\beta=5$ and $\alpha \beta=3$
$\therefore \frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2+\beta^2}{\alpha \beta}=\frac{(\alpha+\beta)^2-2 \alpha \beta}{\alpha \beta}=\frac{(5)^2-2 \times 3}{3}=\frac{19}{3}=$ sum of roots.
Also, $\left(\frac{\alpha}{\beta}\right)\left(\frac{\beta}{\alpha}\right)=1=$ product of roots.
$\therefore$ The quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is
$x^2-\frac{19}{3} x+1=0 \Rightarrow 3 x^2-19 x+3=0$