Question

what is the radius of a circle with the equation ( x^{2}+y^{2}+6x - 2y+3 )?
round your answer to the nearest thousandth.
r =

Explanation:

Step1: Complete the square for \(x\) terms

Group \(x\) terms: \(x^{2}+6x=(x + 3)^{2}-9\)

Step2: Complete the square for \(y\) terms

Group \(y\) terms: \(y^{2}-2y=(y - 1)^{2}-1\)

Step3: Rewrite the circle equation

The original equation \(x^{2}+y^{2}+6x - 2y+3 = 0\) becomes \((x + 3)^{2}-9+(y - 1)^{2}-1+3=0\)
Simplify to \((x + 3)^{2}+(y - 1)^{2}=7\)

Step4: Find the radius

The standard form of a circle equation is \((x - a)^{2}+(y - b)^{2}=r^{2}\). Here \(r^{2}=7\), so \(r=\sqrt{7}\approx2.646\)

Answer:

\(2.646\)