Question

If one root of the quadratic equation $a x^{2}+b x+c=0$ 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

The Correct Option is 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
$\therefore\left(\frac{ c }{ a }\right)^{\frac{1}{n+1}}+\left(\frac{ c }{ a }\right)^{\frac{ n }{n+1}}=-\frac{ b }{ a }$
$\therefore a ^{\frac{n}{n+1}} C ^{\frac{1}{n+1}}+ a ^{\frac{1}{n+1}} C ^{\frac{n}{n+1}}=- b$