Question

determinar o centro e o raio da circunferência cuja equação e 4x*2+4y*2-20x+20y-31=0

Answer 1

$\mathrm{4x^2+4y^2-20x+20y-31=0}\\\\ \mathrm{4x^2-20x+4y^2+20y=31\ \to\ 4(x^2-5x)+4(y^2+5y)=31}\\\\ \mathrm{(x^2-5x)+(y^2+5y)=\dfrac{31}{4}}\\\\ \mathrm{\bigg(x^2-5x+\dfrac{25}{4}\bigg)+\bigg(y^2+5y+\dfrac{25}{4}\bigg)=\dfrac{31}{4}+\dfrac{25}{4}+\dfrac{25}{4}}\\\\ \mathbf{(x-x_c)^2+(y-y_c)^2=r^2\ \to\ \bigg(x-\dfrac{5}{2}\bigg)^2+\bigg(y+\dfrac{5}{2}\bigg)^2=\dfrac{81}{4}}$

$\textbf{An\'alise do centro e do raio da circunfer\^encia:}{}\\\\ \mathrm{\bigg(x-\dfrac{5}{2}\bigg)^2+\bigg(y-\bigg(-\dfrac{5}{2}\bigg)\bigg)^2=\bigg(\dfrac{9}{2}\bigg)^2}\\\\\\ \mathrm{C(x_c,y_c)=C\bigg(\dfrac{5}{2},-\dfrac{5}{2}\bigg)\ \ \bigg\| \ \ r=\dfrac{9}{2}}$