Question

the equation of a circle in the xy - plane is shown above. what is the radius of the circle? a) 2 b) 3 c) 4 d) 9

Explanation:

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

Group \(x\) terms: \(x^{2}+4x=(x + 2)^{2}-4\) (using \((a + b)^{2}=a^{2}+2ab + b^{2}\), here \(a=x\), \(2b = 4\Rightarrow b = 2\))

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

Group \(y\) terms: \(y^{2}-2y=(y - 1)^{2}-1\) (using \((a - b)^{2}=a^{2}-2ab + b^{2}\), here \(a = y\), \(2b=2\Rightarrow b = 1\))

Step3: Rewrite the circle equation

Substitute into \(x^{2}+y^{2}+4x - 2y=-1\):
\((x + 2)^{2}-4+(y - 1)^{2}-1=-1\)
\((x + 2)^{2}+(y - 1)^{2}-5=-1\)
\((x + 2)^{2}+(y - 1)^{2}=4\)
The standard form of a circle equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \(r\) is the radius.

Answer:

A. 2