X^2 + Y^2 + ax + 2y + 3 = 0

What number is this equation the equation of a circle?

Eq: (x+a/2) ² + (y+1) ² = (a²-8) / 4

Center: O (-a/2, - 1)

Radius: r=0.5*sqrt (a²-8)

Mandatory: a>2*sqrt (2)

Step-by-step explanation:

The circle with center in O (xo, yo) and radius r has the equation:

(x-xo) ² + (y-yo) ²=r²

We have:

x²+y²+ax+2y+3=0

But: x²+ax=x²+2 (a/2) x+a²/4-a²/4 = (x+a/2) ²-a²/4

And

y²+2y+3=y²+2y+1+2 = (y+1) ²+2

Replacing, we get:

(x+a/2) ²-a²/4 + (y+1) ²+2=0

(x+a/2) ² + (y+1) ²=a²/4-2 = (a²-8) / 4

By visual inspection we note that:

- center of circle: O (-a/2, - 1)

- radius: r=sqrt ((a²-8) / 4) = 0.5*sqrt (a²-8). This means a²>8 or a>2*sqrt (2)