Question

Ex din poza, va rog! Cred ca trebuie sa faceti patrate perfecte. Dau 90 de puncte ..

Answer 1

$a)\quad x^2+y^2-x-3y+a>0\\ \\ \Big(x-\dfrac{1}{2}\Big)^2+\Big(y-\dfrac{3}{2}\Big)^2-\dfrac{1}{4} - \dfrac{9}{4}+a>0 \\ \\ \Big(x-\dfrac{1}{2}\Big)^2+\Big(y-\dfrac{3}{2}\Big)^2+a-\dfrac{5}{2}>0 \\ \\ \Rightarrow a-\dfrac{5}{2}>0 \Rightarrow a>\dfrac{5}{2} \Rightarrow \boxed{a\in \Big(\dfrac{5}{2},+\infty\Big)}$

$b)\quad 2x^2+3y^2-2x-3y+a+1>0 \\ \\ \Big(\sqrt{2}x-\dfrac{\sqrt 2}{2}\Big)^2+\Big(\sqrt 3y-\dfrac{\sqrt 3}{2}\Big)^2-\dfrac{2}{4}-\dfrac{3}{4}+a+1>0 \\ \\\Big(\sqrt{2}x-\dfrac{\sqrt 2}{2}\Big)^2+\Big(\sqrt 3y-\dfrac{\sqrt 3}{2}\Big)^2+a-\dfrac{1}{4}>0\\ \\\Rightarrow a-\dfrac{1}{4}>0 \Rightarrow a>\dfrac{1}{4}\Rightarrow \boxed{a\in \Big(\dfrac{1}{4},+\infty\Big)}$