Question

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

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

$$\begin{gathered} \Rightarrow x^2-\frac{p}{q}x+\frac{1}{q}=0 \\ \Rightarrow q{x}^2-px+1=0 \end{gathered}$$