Show that x2+y2+Dx+Ey+F=0 represents a circle of positive radius if and only if D2+E2−4F>0 .
Solution:
The equation of the circle can be written as
(x−a)2+(y−b)2=R2 , where (a,b) is a centre of the circle, R is a radius of the circle.
Let's transform our equation to the form of the equation of the circle:
x2+y2+Dx+Ey+F=x2+22Dx+4D2+y2+22Ey+4E2−4D2−4E2+F=0
(x+2D)2+(y+2E)2−4D2−4E2+F=0
(x+2D)2+(y+2E)2=4D2+4E2−F (*)
Compare (*) and the equation of the circle. Equation (*) represents a circle if and only if 4D2+4E2−F=R2 . R2 is always a positive value so we can conclude
4D2+4E2−F>0
or
D2+E2−4F>0
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