Determine the center and the radius of the circle x^2+y^2+10y=0.

The general equation for the family of circles is

with as center and radius

Let us compare our equation with the general equation

there is only one x term, so a becomes 0, i.e.,

but there are two terms of y. To find b value we need to find the roots of the quadratic polynomial of .

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The general equation for the family of circles is

with as center and radius

Let us compare our equation with the general equation

there is only one x term, so a becomes 0, i.e.,

but there are two terms of y. To find b value we need to find the roots of the quadratic polynomial of .

One of the ways to find roots of this polynomial is completing square method.

(equating this to identity)

Now combine and terms and write the equation for the circle

the center of the circle is (0,-5)

and the radius is 5