Question

Aratati ca daca
b) a^2 + b^2 -4a + 10b + 20=0, atunci a>b

Answer 1

$a^2+b^2-4a+10b+20 = 0 \\ a^2-4a+4+b^2+10b+25 - 9 = 0\\ (a-2)^2+(b+5)^2 =9\Leftrightarrow\\ \Leftrightarrow \Big\{a-2=0 si b+5 = 3\Big\} sau \Big\{a-2 = 3 si b+5 = 0\Big\}\\ \\\text{sau} \Big\{a-2 = 0 si b+5 = -3\Big\} sau \Big\{a-2 = -3 si b+5 = 0\Big\} \\ \\ \Leftrightarrow \Big\{a=2 si b=-2\Big\} sau \Big\{a=5 si b=-5\Big\}\\ \\\text{sau} \Big\{a=2 si b=-8\Big\} sau \Big\{a=-1 si b=-5\Big\}$

$\Rightarrow (a,b) = \Big\{(2,-2);(5,-5);(2,-8);(-1,-5)\Big\} \\ \\ 2\ \textgreater \ -2, 5\ \textgreater \ -5,2\ \textgreater \ -8,-1\ \textgreater \ -5\\ \\ \Rightarrow a\ \textgreater \ b$