Saturday, August 12, 2017

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

The general equation for the family of circles is


`(x-a)^2 + (y-b)^2 = r^2 `


with `(a,b) ` as center and `r` radius


Let us compare our equation with the general equation


there is only one x term, so a becomes 0, i.e., `(x-a)^2 = (x-0)^2 = x^2`


but there are two terms of y. To find b value we need to find the roots of the quadratic polynomial of `y^2 + 10 y` .


...

The general equation for the family of circles is


`(x-a)^2 + (y-b)^2 = r^2 `


with `(a,b) ` as center and `r` radius


Let us compare our equation with the general equation


there is only one x term, so a becomes 0, i.e., `(x-a)^2 = (x-0)^2 = x^2`


but there are two terms of y. To find b value we need to find the roots of the quadratic polynomial of `y^2 + 10 y` .


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


`(y)^2 + 2*y*5 + (5)^2 -(5)^2` (equating this to `a^2 + 2 ab + b^2` identity)



`= (y+5)^2 -25 `


Now combine `x` and `y` terms and write the equation for the circle



`(x-0)^2 + [y- (-5)]^2 -25 = 0 `



`(x-0)^2 +[y -(-5)]^2 = 5^2 `


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


and the radius is 5 

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