If `alpha,beta` are the roots of the equation`x^2+p x+1=0;gamma,delta` the roots of the equation `x^2+q x+1=0` , then `(alpha-gamma)(alpha+delta)(beta-gamma)(beta+delta)=`

Options

A. `q^2-p^2`
B. `p^2-q^2`
C. `p^2=q^2`
D. none of these

Answer

A

Explanation

`alpha,beta`
are the roots of the equation`x^2+p x+1=0`

And `gamma,delta`are
the roots of the equation `x^2+q x+1=0`

`implies , alpha + beta = - p, alphabeta = 1 and gamma+delta = - q, gammadelta = 1`

Now `(alpha-gamma)(\alpha+\delta)(beta-gamma)(\beta+\delta)={\alpha \beta-\gamma(\alpha+\beta)+\gamma^2}{alpha \beta+\delta(\alpha+\beta)+\delta^2}`

`\Rightarrow(\alpha -\gamma )(\alpha+\delta)(\beta-\gamma)(\beta+\delta)={\alpha \beta-\gamma(\alpha+\beta)+\gamma^2}{\alpha \beta+\delta(\alpha+\beta)+\delta^2}`

`\Rightarrow (alpha-gamma)(alpha+delta)(beta-gamma)(beta+delta)={1+\gamma p+\gamma^2}+{1-\delta p+\delta^2))`

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