Question

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

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

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 as $\alpha ,\beta$
To find: Transformed quadratic equation with roots as $\frac{1}{\alpha }\&\frac{1}{\beta }$

Let $t=\frac{1}{x}$
$\because x=\alpha \Rightarrow t=\frac{1}{\alpha }$
$\&x=\beta \Rightarrow t=\frac{1}{\beta }$

Replacing $x\to \frac{1}{t},$ we get:

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

In terms of variable $x,$ we get the equation as:

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