Question

If $\alpha$and $\beta$ are the roots of the equation$x^2+px+2=0$ and$(1/\alpha )$ and $(1/\beta )$are the roots of the equation$2x^2+2qx+1=0$, then$\left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)\left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)$ is equal to

• A. $\frac{9}{4}(9+p^2)$
• B. $\frac{9}{4}(9+q^2)$
• C. $\frac{9}{4}(9-p^2)$
• D. $\frac{9}{4}(9-q^2)$

Answer 1

The correct option is C

$\frac{9}{4}(9-p^2)$

Explanation for the correct option:

Step 1. Find the value of $\left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)$:

Given that $\alpha$and $\beta$ are the roots of the equation$x^2+px+2=0$ .

$\Rightarrow (\alpha +\beta )=-p;\alpha .\beta =2$

And also given $(1/\alpha )$ and $(1/\beta )$are the roots of the equation$2x^2+2qx+1=0$,

$\Rightarrow \frac{1}{\alpha }+\frac{1}{\beta }=\frac{-2q}{2}=-q;\left(\frac{1}{\alpha }\right)\left(\frac{1}{\beta }\right)=\frac{1}{2}$

$$\begin{gathered} \left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)=\alpha \beta +\frac{1}{\alpha \beta }+2 \\ \Rightarrow \left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)=2+\frac{1}{2}+2 \\ \Rightarrow \left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)=\frac{9}{2}..........\left(i\right) \end{gathered}$$

Step 2. Find the value of $\mathbf{(}\mathbf{\alpha }\mathbf{-}\frac{\mathbf{1}}{\mathbf{\alpha }}\mathbf{)}\mathbf{(}\mathbf{\beta }\mathbf{-}\frac{\mathbf{1}}{\mathbf{\beta }}\mathbf{)}$:

$$\begin{gathered} \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\alpha \beta +\frac{1}{\alpha \beta }-\frac{\alpha }{\beta }-\frac{\beta }{\alpha } \\ \Rightarrow \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=2+\frac{1}{2}-\frac{({\alpha }^2+{\beta }^2)}{\alpha \beta } \\ \Rightarrow \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{5}{2}-\frac{({\alpha }^2+{\beta }^2+2\alpha \beta -2\alpha \beta )}{\alpha \beta } \\ \Rightarrow \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{5}{2}-\frac{{(\alpha +\beta )}^2-2\alpha \beta }{\alpha \beta } \\ \Rightarrow \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{5}{2}-\frac{p^2-4}{2} \\ \Rightarrow \left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{(9-p^2)}{2}..........\left(ii\right) \end{gathered}$$

Multiplying equations (i) and (ii). we get

$$\begin{gathered} \left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)\left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{9}{2}\left(\frac{9-p^2}{2}\right) \\ \Rightarrow \left(\alpha +\frac{1}{\beta }\right)\left(\beta +\frac{1}{\alpha }\right)\left(\alpha -\frac{1}{\alpha }\right)\left(\beta -\frac{1}{\beta }\right)=\frac{9}{4}(9-p^2) \end{gathered}$$

Hence, the correct option is (C).