Question:
If the difference of the roots of $x^{2}-p x+q=0$ is unity, then
(a) $p^{2}+4 q=1$
(b) $p^{2}-4 q=1$
(c) $p^{2}+4 q^{2}=(1+2 q)^{2}$
(d) $4 p^{2}+q^{2}=(1+2 p)^{2}$
Solution:
(b) $p^{2}-4 q=1$
Given equation: $x^{2}-p x+q=0$
Also $\alpha$ and $\beta$ are the roots of the equation such that $\alpha-\beta=1$.
Sum of the roots $=\alpha+\beta=\frac{-\text { Coefficient of } x}{\text { Coefficient of } x^{2}}=-\left(\frac{-p}{1}\right)=p$
Product of the roots $=\alpha \beta=\frac{C \text { onstant term }}{C \text { oefficient of } x^{2}}=q$
$\therefore(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$
$\Rightarrow p^{2}-1=4 q$
$\Rightarrow p^{2}-4 q=1$