[tex]\displaystyle \sf \text{Equa\c c\~ao reduzida da circunfer\^encia} : \\\\ (x-x_c)^2+(y-y_c)^2=R^2 \\\\ \text{centro : }(x_c\ ,\ y_c) \\\\ raio : R \\\\\\\\ temos : \\\\\ 2x^2+2y^2+4x-y+k=0 \\\\ \text{circunfer\^encia de raio 1. }\\\\ \text{colocando na forma reduzida} : \\\\\ 2x^2+2y^2+4x-y+k=0\ \ (\div 2) \\\\ \frac{2x^2}{2}+\frac{2y^2}{2}+\frac{4x}{2}-\frac{y}{2}+\frac{k}{2}=0 \\\\ x^2+2x+y^2-y+\frac{k}{2} = 0 \\\\\ \text{somando 1 e 1/16 em ambos os lados da igualdade } :[/tex]
[tex]\displaystyle \sf \left(x^2+2x+1\right) +\left(y^2-\frac{y}{2}+\frac{1}{16}\right)+\frac{k}{2}=1+\frac{1}{16} \\\\\\ (x+1)^2+\left(y-\frac{1}{4}\right)^2=\frac{1\cdot 16+1}{16}-\frac{k}{2} \\\\\\ (x+1)^2+\left(y-\frac{1}{4}\right)^2 =\frac{17}{16}-\frac{8k}{16} \\\\\\ \text{ent\~ao temos}: \\\\ \frac{17-8k}{16} = R^2 \\\\\\ \frac{17-8k}{16}=1^2=1 \\\\\ 17-8k=16 \\\\ 8k = 17-16 \\\\ 8k = 1 \\\\ \large\boxed{\sf \ k = \frac{1}{8}\ }\checkmark[/tex]
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[tex]\displaystyle \sf \text{Equa\c c\~ao reduzida da circunfer\^encia} : \\\\ (x-x_c)^2+(y-y_c)^2=R^2 \\\\ \text{centro : }(x_c\ ,\ y_c) \\\\ raio : R \\\\\\\\ temos : \\\\\ 2x^2+2y^2+4x-y+k=0 \\\\ \text{circunfer\^encia de raio 1. }\\\\ \text{colocando na forma reduzida} : \\\\\ 2x^2+2y^2+4x-y+k=0\ \ (\div 2) \\\\ \frac{2x^2}{2}+\frac{2y^2}{2}+\frac{4x}{2}-\frac{y}{2}+\frac{k}{2}=0 \\\\ x^2+2x+y^2-y+\frac{k}{2} = 0 \\\\\ \text{somando 1 e 1/16 em ambos os lados da igualdade } :[/tex]
[tex]\displaystyle \sf \left(x^2+2x+1\right) +\left(y^2-\frac{y}{2}+\frac{1}{16}\right)+\frac{k}{2}=1+\frac{1}{16} \\\\\\ (x+1)^2+\left(y-\frac{1}{4}\right)^2=\frac{1\cdot 16+1}{16}-\frac{k}{2} \\\\\\ (x+1)^2+\left(y-\frac{1}{4}\right)^2 =\frac{17}{16}-\frac{8k}{16} \\\\\\ \text{ent\~ao temos}: \\\\ \frac{17-8k}{16} = R^2 \\\\\\ \frac{17-8k}{16}=1^2=1 \\\\\ 17-8k=16 \\\\ 8k = 17-16 \\\\ 8k = 1 \\\\ \large\boxed{\sf \ k = \frac{1}{8}\ }\checkmark[/tex]