Question

If $\alpha \&\beta$ are the roots of equation. $x^2+px+q=0$, then - $-\frac{1}{\alpha },-\frac{1}{\beta }$ are the roots of the equation.

• A. $x^2-px+q=0$
• B. $x^2+px+q=0$
• C. $q{x}^2+px+1=0$
• D. $q{x}^2-px+1=0$

Answer 1

The correct option is D

$q{x}^2-px+1=0$

Given equation : $x^2+px+q=0$ Also, $\alpha \&\beta$ are the roots of the given equation. Then, sum of the roots = $\alpha +\beta$ = -p Now, for roots - $\frac{1}{\alpha },-\frac{1}{\beta }$, we have : Sum of the roots = - $\frac{1}{\alpha },-\frac{1}{\beta }=-\frac{\alpha +\beta }{\alpha \beta }$ Hence, the equation involving the roots $=-\left(\frac{-p}{q}\right)=\frac{p}{q}$ $\text{Hence, the equation involving the roots}-\frac{1}{\alpha },-\frac{1}{\beta }isasfollows:x^2-(\alpha +\beta )x+\alpha \beta =0\Rightarrow x^2-\frac{p}{q}x+\frac{1}{q}=0\Rightarrow q{x}^2-px+1=0$