question_answer

If \[|z-4+3i|\le 1\]  and \[\alpha \] and \[\beta \] be the greatest and least value of |z| respectively and K be the least value of \[\frac{{{x}^{4}}+{{x}^{2}}+4}{x}\] on the interval \[\left( 0,\,\infty  \right)\] then K is equal to

A)  \[\alpha \]                                         

B)  \[\beta \]

C)  \[\alpha +\beta \]                          

D)  \[\alpha -\beta \]

Correct Answer: A

Solution :

\[\because \,\,|Z-(4-3i)|\le 1\] \[|Z{{|}_{\max }}=OC+AC=5+1=6=\alpha \] \[|Z{{|}_{\min }}=OC-BC=5-1=4=\beta \] Also, get \[y=\frac{{{x}^{4}}+{{x}^{2}}+4}{x}={{x}^{3}}+x+\frac{4}{x}\] \[={{x}^{3}}+x+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}\] For \[x\in (0,\,\infty )\,AM\ge GM\] \[\frac{{{x}^{3}}+x+\frac{4}{x}}{6}\ge \sqrt{1}\] \[\Rightarrow \,\,{{x}^{3}}+x+\frac{4}{x}\ge 6=k=\alpha \]