Question

If the difference between the roots of $x^2+ax-b=0$ is equal to the difference between the roots of $x^2-px+q=0,$ then $a^2-p^2$ in terms of $b$ and $q$ is

• A. $-4(b+q)$
• B. $4(b+q)$
• C. $4(b-q)$
• D. $4(q-b)$

Answer 1

Answer: (A)

Let $\alpha ,\beta$ are the roots of the equation
$x^2+ax-b=0$
$\therefore$ $\alpha +\beta =-a,\alpha \beta =-b$
and $\gamma ,\delta$
are the roots of the equation
$x^2-px+q=0$
$\therefore$ $\gamma +\delta =p,\gamma \delta =q$
Given, $\alpha -\beta =\gamma -\delta$
$\Rightarrow$ ${(\alpha -\beta )}^2={(\gamma -\delta )}^2$
$\Rightarrow$ ${(\alpha +\beta )}^2-4\alpha \beta ={(\gamma +\delta )}^2-4\gamma \delta$
$\Rightarrow$ $a^2+4b=p^2-4q$
$\Rightarrow$ $a^2-p^2=-4(b+q)$