ISI2016-DCG-28

Question

If one root of a quadratic equation $ax^{2}+bx+c=0$ be equal to the n th power of the other, then

1. $(ac)^{\frac{n}{n+1}}+b=0$
2. $(ac)^{\frac{n+1}{n}}+b=0$
3. $(ac^{n})^{\frac{1}{n+1}}+(a^{n}c)^{\frac{1}{n+1}}+b=0$
4. $(ac^\frac{1}{n+1})^{n}+(a^\frac{1}{n+1}c)^{n+1}+b=0$

Answer 1

$\underline{\mathbf{Answer:3}}$

$\underline{\mathbf{Solution:}}$

Let $\alpha$ and $\beta$ are the roots of the equation $\mathrm{ax^2 + bx + c = 0}$

then

$\mathrm {B = \alpha^n}\;\text{[Given]}$

Also we know that:

$\text{Sum of roots} = -\dfrac{b}{a}$

$\text{Product of roots} = \mathrm{\dfrac{c}{a}}$

$\therefore \alpha = \mathrm{\frac{c}{a}}\Rightarrow \alpha\times \alpha^n =\alpha^{n+1}= \dfrac{c}{a} \Rightarrow \alpha = (\frac{c}{a})^{\frac{1}{n+1}}$

then,

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

Now we can write the quadratic equation $\mathrm{ax^2+bx+c = 0}$ using one of the root $\alpha$ as:

$\mathrm{a\alpha^2 + b\alpha + c = 0}$

$\Rightarrow \mathrm{a\alpha +b + \frac{c}{\alpha}}$

$\mathrm{\Rightarrow \frac{c}{\alpha} + a\alpha + b = 0}$

$\mathrm{\dfrac{c}{(\frac{c}{a})^{\frac{1}{n+1}}}+a\left (\frac{c}{a} \right )^{\frac{1}{n+1}}+b = 0}$

$\mathrm{c \left (\frac{a}{c} \right )^{\frac{1}{n+1}}+a\left ( \frac{c}{a}\right)^{\frac{1}{n+1}}+b = 0}$

$\mathrm{\left ( \frac{c^{n+1}a}{c}\right)^{\frac{1}{n+1}}+ \left ( \frac{a^{n+1}c}{a}\right)^{\frac{1}{n+1}} + b = 0}$

$\mathrm{\Rightarrow (ac^n)^{\frac{1}{n+1}} + (a^nc)^{\frac{1}{n+1}} + b = 0}$

$\therefore \;\mathrm{Option}\; 3$ is the correct answer.

Comments

Yeah, you are correct.

That's a typing error.

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PS

Corrected!

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Yeah. It's now correct.