SOLUTION: Determine whether the equation represents a circle, a point, or has no graph. If the equation is that of a circle, find its center and radius. (2x^2)+(2y^2)-2x+8y=1/2

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Determine whether the equation represents a circle, a point, or has no graph. If the equation is that of a circle, find its center and radius. (2x^2)+(2y^2)-2x+8y=1/2       Log On


   

Question 1008532: Determine whether the equation represents a circle, a point, or has no graph. If the equation is that of a circle, find its center and radius.
(2x^2)+(2y^2)-2x+8y=1/2

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
the formula of the circle is:
%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2
where h and k are the x and y coordinates of the center of the circle

2x%5E2%2B2y%5E2-2x%2B8y=1%2F2+......both sides divide by 2
x%5E2%2By%5E2-x%2B4y=1%2F4+
%28x%5E2-x%29%2B%28y%5E2%2B4y%29=1%2F4+......complete square
%28x%5E2-x%2Bb%5E2%29-b%5E2%2B%28y%5E2%2B4y%2Bb%5E2%29-b%5E2=1%2F4+....recall that %28a%2B-b%29%5E2=a%5E2%2B2ab%2Bb%5E2 or a%5E2-2ab%2Bb%5E2
%28x%5E2-x%2Bb%5E2%29-b%5E2%2B%28y%5E2%2B4y%2Bb%5E2%29-b%5E2=1%2F4+...so, in your case a=1, and 2ab=1 in x part, and a=1, and 2ab=4 in y part
then,
2ab=1=>2%2A1%2Ab=1b=1%2F2 in x part
and 2ab=4=>2%2A1%2Ab=4=>b=4%2F2=> b=2 in y part

so, we have

%28x%5E2-x%2B%281%2F2%29%5E2%29-%281%2F2%29%5E2%2B%28y%5E2%2B4y%2B2%5E2%29-2%5E2=1%2F4+

%28x-1%2F2%29%5E2-%281%2F4%29%2B%28y%2B2%29%5E2-4=1%2F4+

%28x-1%2F2%29%5E2%2B%28y%2B2%29%5E2=1%2F4%2B%281%2F4%29%2B4+

%28x-1%2F2%29%5E2%2B%28y%2B2%29%5E2=2%2F4%2B4+
%28x-1%2F2%29%5E2%2B%28y%2B2%29%5E2=1%2F2%2B4+
%28x-1%2F2%29%5E2%2B%28y%2B2%29%5E2=1%2F2%2B8%2F2+
%28x-1%2F2%29%5E2%2B%28y%2B2%29%5E2=9%2F2+
so, it is a circle and
h=1%2F2
k=-2
r=sqrt%289%2F2%29=sqrt%289%29%2Fsqrt%282%29=3%2Fsqrt%282%29
the center is at:(1%2F2,-2) and radius is r=3%2Fsqrt%282%29