Question

what is the radius of the circle ( x^2 + (y + 1)^2 = 1 )? write your answer in simplified, rationalized form.

Explanation:

Step1: Recall the standard circle equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Compare with the given equation

The given equation is \(x^2 + (y + 1)^2 = 1\). We can rewrite \(x^2\) as \((x - 0)^2\) and \((y + 1)^2\) as \((y - (-1))^2\). So comparing with \((x - h)^2 + (y - k)^2 = r^2\), we have \(r^2 = 1\).

Step3: Solve for \(r\)

To find \(r\), we take the square root of both sides of \(r^2 = 1\). Since radius is a non - negative quantity, \(r=\sqrt{1}=1\).

Answer:

1