Question

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

• A. $\frac{x^2}{a}+\frac{x}{b}+\frac{1}{c}=0$
• B. $c{x}^2+bx+a=0$
• C. $b{x}^2+cx+a=0$
• D. $a{x}^2+cx+b=0$

Answer 1

The correct option is B

$c{x}^2+bx+a=0$

Explanation for the correct option:

Step 1. Find the quadratic equation:

As we know,

$\alpha +\beta =\frac{–b}{a}$ and $\alpha \beta =\frac{c}{a}$

The given roots of the equation are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$

Step 2. Sum of the roots is:

$\begin{array}{rcl}& \Rightarrow & \frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }\\ & =& \frac{–b}{c}\end{array}$

Step 3. Product of the roots is:

$\frac{1}{\alpha }·\frac{1}{\beta }=\frac{a}{c}$

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

$\Rightarrow$ $c{x}^2+bx+a=0$

Thus, the required quadratic equation is $\mathit{c}{\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Hence, Option ‘B’ is Correct.