Question

Identify the center and the radius of the circle with the equation 〖2x〗^2+〖2y〗^2+10x=2y+7 in each item. Sketch its graph, and indicate the center.

Answer 1

$\left(x−a\right)^2+\left(y−b\right)^2=r^2\:\:\mathrm{is\:the\:circle\:equation\:with\:a\:radius\:r,\:centered\:at}\:\left(a,\:b\right)$

Rewriting $2x^2+2y^2+10x=2y+7$ in form of standard circle equation

$2x^2+10x+2y^2-2y=7$

$x^2+5x+y^2-y=\frac{7}{2}$

Writing the above equation in form of $\left(x−a\right)^2+\left(y−b\right)^2=r^2$

$\left(x-\left(-\frac{5}{2}\right)\right)^2+\left(y-\frac{1}{2}\right)^2=\left(\sqrt{10}\right)^2$

$\left(a,\:b\right)=\left(-\frac{5}{2},\:\frac{1}{2}\right),\:r=\sqrt{10}$

$\mathrm{Circle\:with\:center\:at}\:\left(-\frac{5}{2},\:\frac{1}{2}\right)\:\mathrm{and\:radius}\:r=\sqrt{10}$

Sketch its graph is below