Question Number 157947 by cortano last updated on 30/Oct/21
$${What}\:{are}\:{the}\:{coordinates}\:{of}\:{the} \\ $$$${points}\:{on}\:{the}\:{curve}\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} =\mathrm{16} \\ $$$${which}\:{nearest}\:{to}\:\left(\mathrm{0},\mathrm{6}\right)? \\ $$
Answered by mr W last updated on 30/Oct/21
$${d}^{\mathrm{2}} ={x}^{\mathrm{2}} +\left({y}−\mathrm{6}\right)^{\mathrm{2}} \\ $$$$=\mathrm{16}+{y}^{\mathrm{2}} +\left({y}−\mathrm{6}\right)^{\mathrm{2}} \\ $$$$=\mathrm{2}\left({y}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{34}\geqslant\mathrm{34} \\ $$$${d}\geqslant\sqrt{\mathrm{34}}={nearest}\:{distance} \\ $$$${at}\:{y}=\mathrm{3},\:{x}=\pm\sqrt{\mathrm{16}+\mathrm{3}^{\mathrm{2}} }=\pm\mathrm{5} \\ $$
Commented by cortano last updated on 30/Oct/21
$${yes} \\ $$