1. Tardigrade
  2. Question
  3. Mathematics
  4. Let k be a real number such that k ≠ 0. If α and β are non zero complex numbers satisfying α+β=-2 k and α2+β2=4 k2-2 k, then a quadratic equation having (α+β/α) and (α+β/β) as its roots is equal to

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Q. Let $k$ be a real number such that $k \neq 0$. If $\alpha$ and $\beta$ are non zero complex numbers satisfying $\alpha+\beta=-2 k$ and $\alpha^2+\beta^2=4 k^2-2 k$, then a quadratic equation having $\frac{\alpha+\beta}{\alpha}$ and $\frac{\alpha+\beta}{\beta}$ as its roots is equal to

Complex Numbers and Quadratic Equations

Solution:

We have $\alpha+\beta=-2 k$
Also, $ \alpha^2+\beta^2=4 k ^2-2 k \Rightarrow(\alpha+\beta)^2-2 \alpha \beta=4 k ^2-2 k \Rightarrow \alpha \beta= k$
$\therefore $ Sum of roots $=\frac{\alpha+\beta}{\alpha}+\frac{\alpha+\beta}{\beta}=\frac{(\alpha+\beta)^2}{\alpha \beta}=\frac{4 k ^2}{ k }=4 k$
and product of roots $=\left(\frac{\alpha+\beta}{\alpha}\right)\left(\frac{\alpha+\beta}{\beta}\right)=\frac{4 k ^2}{ k }=4 k$
Hence required quadratic equation is $x ^2-4 kx +4 k =0 $