• # question_answer If a, b be the roots of ${{x}^{2}}+px+q=0$ and $\alpha +h,\,\beta +h$ are the roots of ${{x}^{2}}+rx+s=0$, then [AMU 2001] A) $\frac{p}{r}=\frac{q}{s}$ B) $2h=\left[ \frac{p}{q}+\frac{r}{s} \right]$ C) ${{p}^{2}}-4q={{r}^{2}}-4s$ D) $p{{r}^{2}}=q{{s}^{2}}$

$\alpha +\beta =-p,\,\alpha \beta =q$ $\alpha +\beta +2h=-r,$ $(\alpha +h)(\beta +h)=s$ $-p+2h=-r\Rightarrow h=\frac{p-r}{2}$ ?..(i) Now, $\alpha \beta +h\,(\alpha +\beta )+{{h}^{2}}=s$ $\Rightarrow q+h(-p)+{{h}^{2}}=s$ $\Rightarrow \,\,q+\left( \frac{p-r}{2} \right)\,(-p)+{{\left( \frac{p-r}{2} \right)}^{2}}=s$ $\Rightarrow q-\frac{({{p}^{2}}-pr)}{2}+\frac{{{p}^{2}}+{{r}^{2}}-2pr}{4}=s$ $\Rightarrow 4q-2{{p}^{2}}+2pr+{{p}^{2}}+{{r}^{2}}-2pr=4s$ $\Rightarrow 4q-{{p}^{2}}+{{r}^{2}}-4s=0$ $\Rightarrow {{r}^{2}}-4s={{p}^{2}}-4q$.