Question

If $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,a\ne 0$. Find the equation with roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }.$

• A. ${\left(\frac{a}{x}\right)}^2+\left(\frac{b}{x}\right)+c=0$
• B. $a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+c=0$
• C. $a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+cx=0$
• D. None of the above

Answer 1

The correct option is B $a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+c=0$

Given: $a{x}^2+bx+c=0,a\ne 0$ with roots $\alpha ,\beta$
To find: Transformed quadratic equation with roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$

Let,$t=\frac{1}{x}$

Since, $x=\alpha \Rightarrow t=\frac{1}{\alpha }$
$x=\beta \Rightarrow t=\frac{1}{\beta }$

Replace $x\to \frac{1}{t}$

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

In terms of variable x,

$a{\left(\frac{1}{x}\right)}^2+b\left(\frac{1}{x}\right)+c=0$
This is required equation.