Theory of Equations 1 Question 52
53. If one root of the quadratic equation $a x^{2}+b x+c=0$ is equal to the $n$th power of the other, then show that
$$ \left(a c^{n}\right)^{\frac{1}{n+1}}+\left(a^{n} c\right)^{\frac{1}{n+1}}+b=0 $$
(1983, 2M)
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Solution:
- Let $\alpha, \beta$ are roots of $a x^{2}+b x+c=0$
Given, $\quad \alpha=\beta^{n}$
$$ \begin{aligned} \Rightarrow & \alpha \beta=\frac{c}{a} \Rightarrow \beta^{n+1}=\frac{c}{a} \\ \Rightarrow & \beta=\frac{c}{a}^{1 /(n+1)} \end{aligned} $$
It must satisfy $a x^{2}+b x+c=0$
$$ \begin{aligned} & \text { i.e. } \quad a \frac{c}{a}^{2 /(n+1)}+b \frac{c}{a}^{1 /(n+1)}+c=0 \\ & \Rightarrow \quad \frac{a \cdot c^{2 /(n+1)}}{a^{2 /(n+1)}}+\frac{b \cdot c^{1 /(n+1)}}{a^{1 /(n+1)}}+c=0 \\ & \Rightarrow \quad \frac{c^{1 /(n+1)}}{a^{1 /(n+1)}} \quad \frac{a \cdot c^{1 /(n+1)}}{a^{1 /(n+1)}}+b+\frac{c \cdot a^{1 /(n+1)}}{c^{1 /(n+1)}}=0 \\ & \Rightarrow \quad a^{n /(n+1)} c^{1 /(n+1)}+b+c^{n /(n+1)} a^{1 /(n+1)}=0 \\ & \Rightarrow \quad\left(a^{n} c\right)^{1 /(n+1)}+\left(c^{n} a\right)^{1 /(n+1)}+b=0 \end{aligned} $$
