Question Number 148991 by gsk2684 last updated on 02/Aug/21
$${The}\:{largest}\:{value}\:{of}\:{k}\:{for}\:{which}\: \\ $$$${the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={k}^{\mathrm{2}} \:{lies}\:{completely} \\ $$$${in}\:{the}\:{interior}\:{of}\:{the}\:{parabola} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{x}+\mathrm{16}\:? \\ $$
Answered by mr W last updated on 02/Aug/21
$${x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{16}−{k}^{\mathrm{2}} =\mathrm{0} \\ $$$$\Delta=\mathrm{4}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{16}−{k}^{\mathrm{2}} \right)\leqslant\mathrm{0} \\ $$$${k}^{\mathrm{2}} \leqslant\mathrm{12} \\ $$$$\Rightarrow−\mathrm{2}\sqrt{\mathrm{3}}\leqslant{k}\leqslant\mathrm{2}\sqrt{\mathrm{3}} \\ $$