Question

If one root of the quadratic equation $a{x}^2+bx+c=0$ is equal to $n^{th}$ 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)

$-b$

Let one root be $\alpha$

Then the other root is ${\alpha }^n$

So,product of roots $=\frac{c}{a}$

$\therefore \left(\alpha \right)\left({\alpha }^n\right)=\frac{c}{a}$

$\therefore {\alpha }^{n+1}=\frac{c}{a}$

$\therefore \alpha =(\frac{c}{a}{)}^{\frac{1}{n+1}}$ ...(1)

sum of roots $=-\frac{b}{a}$

$\therefore \alpha +{\alpha }^n=-\frac{b}{a}$

Substituting the value of $\alpha$ from equation (1), we get

$\therefore (\frac{c}{a}{)}^{\frac{1}{n+1}}+(\frac{c}{a}{)}^{\frac{n}{n+1}}=-\frac{b}{a}$

$\therefore a^{\frac{n}{n+1}}{c}^{\frac{1}{n+1}}+a^{\frac{1}{n+1}}{c}^{\frac{n}{n+1}}=-b$

Answer is option (B)