Question

If, $\alpha$ and $\beta$ are the positive roots of the equation, $a{x}^2+bx+c=0$, find the equation whose roots are $\frac{1}{\sqrt{\alpha }}$ and $\frac{1}{\sqrt{\beta }}.$

• A. $a{x}^4+b{x}^2+c=0$
• B. None of the above
• C. $c{x}^2+bx+a=0$
• D. $c{x}^4+b{x}^2+a=0$

Answer 1

The correct option is D $c{x}^4+b{x}^2+a=0$

Given: $\alpha ,\beta$ are the positive roots of the equation $a{x}^2+bx+c=0$
To find: Equation with roots $\frac{1}{\sqrt{\alpha }}$ and $\frac{1}{\sqrt{\beta }}$

Let $y=\frac{1}{\sqrt{x}}$

$\Rightarrow x=\frac{1}{y^2}$

Put $x=\frac{1}{y^2}$ in given equation,

$a{\left(\frac{1}{y^2}\right)}^2+b\left(\frac{1}{y^2}\right)+c=0$

$\Rightarrow \frac{a}{y^4}+\frac{b}{y^2}+c=0$

$\Rightarrow c{y}^4+b{y}^2+a=0$

Now, In terms of variables $x,$ the transformed equation becomes:

$c{x}^4+b{x}^2+a=0$