Question

If $\alpha ,\beta$ are roots of the equation $x^2+px-q=0$ and $\gamma ,\delta$ are roots of $x^2+px+r=0,$ then the value of $(\alpha -\gamma )(\alpha -\delta )$ is-

• A. p+r
• B. p-r
• C. q-r
• D. q+r

Answer 1

The correct option is D q+r

We have,

$\alpha and\beta$ are the roots of the equation,

$x^2+px-q=0$

Then,

Sum of roots $\alpha +\beta =-\frac{coeff.ofx}{coeff.of{x}^2}=-\frac{p}{1}=-p$

Product of roots $\alpha .\beta =\frac{\text{constant}\text{term}}{\text{coeff}\text{.}\text{of}{\text{x}}^{\text{2}}}=\frac{-q}{1}=-q$

Now,

$\gamma and\delta$ be the roots of the equation,

$x^2+px+r=0$

So,

Sum of roots $\gamma +\delta =-\frac{coeff.ofx}{coeff.of{x}^2}=-\frac{p}{1}=-p$

Product of roots $\gamma .\delta =\frac{\text{constant}\text{term}}{\text{coeff}\text{.}\text{of}{\text{x}}^{\text{2}}}=\frac{r}{1}=r$

Then,

$(\alpha -\gamma )(\alpha -\delta )$

$\alpha^2-\alpha\delta-\alpha\gamma+\delta\gamma$

${\alpha }^2-\alpha (-p)+r$

${\alpha }^2+pq+r$

$q+r$

Hence, this is the answer.