Question

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to $n^{\text{th }}$ power of the other root, then the value of: $a^{\frac{n}{n-1}}{C}^{\frac{1}{n-1}}+c^{\frac{n}{n-1}}{a}^{\frac{1}{n-1}}$ is equal to

• A. $b$
• B. $-b$
• C. $\frac{1}{b^{n+1}}$
• D. $-b^{n+1}$

Answer 1

Answer: (B)

Let one root be $\alpha$
Then the other root is ${\alpha }^n$
So,product of roots $=\frac{c}{a}$
$\therefore (\alpha )\left({\alpha }^n\right)=\frac{c}{a}$
$\therefore {\alpha }^{n+1}=\frac{c}{a}$
$\therefore \alpha ={\left(\frac{c}{a}\right)}^{\frac{1}{n+1}}$...(1)
sum of roots $=-\frac{b}{a}$
$\therefore \alpha +{\alpha }^n=-\frac{b}{a}$
Substituting the value of $\alpha$ from equation (1), we get
$\therefore {\left(\frac{c}{a}\right)}^{\frac{1}{n+1}}+{\left(\frac{c}{a}\right)}^{\frac{n}{n+1}}=-\frac{b}{a}$
$\therefore a^{\frac{n}{n+1}}{C}^{\frac{1}{n+1}}+a^{\frac{1}{n+1}}{C}^{\frac{n}{n+1}}=-b$