Question #326808

Show that X^2 + Y^2 + Dx + Ey + F=0 represents a circle of positive radius if and only if D^2 + E^2 - 4F > 0.

1
Expert's answer
2022-04-12T12:35:46-0400

Show that x2+y2+Dx+Ey+F=0x^2+y^2+Dx+Ey+F=0 represents a circle of positive radius if and only if D2+E24F>0D^2 + E^2 - 4F > 0 .

Solution:

The equation of the circle can be written as

(xa)2+(yb)2=R2(x-a)^2+(y-b)^2=R^2 , where (a,b)(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+2D2x+D24+x^2+y^2+Dx+Ey+F=x^2+2\frac D2x+\frac{D^2}{4}+y2+2E2y+E24D24E24+F=0y^2+2\frac E2y+\frac{E^2}{4}-\frac{D^2}{4}-\frac{E^2}{4}+F=0

(x+D2)2+(y+E2)2D24E24+F=0(x+\frac D2)^2+(y+\frac E2)^2-\frac{D^2}{4}-\frac{E^2}{4}+F=0

(x+D2)2+(y+E2)2=D24+E24F(x+\frac D2)^2+(y+\frac E2)^2=\frac{D^2}{4}+\frac{E^2}{4}-F (*)

Compare (*) and the equation of the circle. Equation (*) represents a circle if and only if D24+E24F=R2\frac{D^2}{4}+\frac{E^2}{4}-F=R^2 . R2R^2 is always a positive value so we can conclude

D24+E24F>0\frac{D^2}{4}+\frac{E^2}{4}-F>0

or

D2+E24F>0D^2+E^2-4F>0


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