SOLUTION: Complete the squares in the general equation x^2+ax+y^2+by+c=0 and simplify tge result as much as possible. Under what conditions on the coefficient a,b and c does this equation re

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Complete the squares in the general equation x^2+ax+y^2+by+c=0 and simplify tge result as much as possible. Under what conditions on the coefficient a,b and c does this equation re      Log On


   

Question 1144744: Complete the squares in the general equation x^2+ax+y^2+by+c=0 and simplify tge result as much as possible. Under what conditions on the coefficient a,b and c does this equation represent a circle? A point? The empty set? In case that the equation does represents a circle find it's center and radius.
Answer by KMST(5431) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2Bax%2By%5E2%2Bby%2Bc=0 <--> highlight%28x%5E2%2Bax%2By%5E2%2Bby=-c%29
Examining the terms in each variable, we find
x%5E2%2Bax as the terms in x , and
y%5E2%2Bby as the terms in y .
Those two expressions would be part of the squares
x%5E2%2Bax%2B%28a%2F2%29%5E2=%28x%2Ba%2F2%29%5E2 , and
y%5E2%2Bby%2B%28b%2F2%29%5E2=%28y%2Bb%2F2%29%5E2 .
We can complete those squares by adding
%28a%2F2%29%5E2%2B%28b%2F2%29%5E2=a%5E2%2F4%2Bb%5E2%2F4
to the left and right sides of the equation highlighted above to get
x%5E2%2Bax%2B%28a%2F2%29%5E2%2By%5E2%2Bby%2B%28b%2F2%29%5E2%22=%22a%5E2%2F4%22%2B%22b%5E2%2F4%22-%22c
%28x%5E2%2Bax%2B%28a%2F2%29%5E2%29%2B%28y%5E2%2Bby%2B%28b%2F2%29%5E2%29%22=%22%28a%5E2%2Bb%5E2-4c%29%2F4
highlight%28%28x%2Ba%2F2%29%5E2%2B%28y%2Bb%2F2%29%5E2=%28a%5E2%2Bb%5E2-4c%29%2F4%29

If a%5E2%2Bb%5E2-4c%3E0 <--> %28a%5E2%2Bb%5E2-4c%29%2F4%3E0 , we can say
%28a%5E2%2Bb%5E2-4c%29%2F4%22=%22R%5E2 ,
and write the equation as highlight%28%28x%2Ba%2F2%29%5E2%2B%28y%2Bb%2F2%29%5E2=R%5E2%29 , which represents a circle of radius R , centered at O%28-4%2F2%2C-b%2F2%29 .

If a%5E2%2Bb%5E2-4c=0 <--> %28a%5E2%2Bb%5E2-4c%29%2F4=0 , we can write the equation as highlight%28%28x%2Ba%2F2%29%5E2%2B%28y%2Bb%2F2%29%5E2=0%29 --> system%28x%2Ba%2F2=0%2Cy%2Bb%2F2=0%29 --> highlight%28system%28x=-a%2F2%2Cy=-b%2F2%29%29
which represents the point O%28-4%2F2%2C-b%2F2%29 .

If a%5E2%2Bb%5E2-4c%3C0 <--> %28a%5E2%2Bb%5E2-4c%29%2F4%3C0 ,
the equation turns into the inequality highlight%28%28x%2Ba%2F2%29%5E2%2B%28y%2Bb%2F2%29%5E2%3C0%29 , which has no real solutions and represents the empty set of (x,y) points.