Finding the Radius of a Circle from its Equation

The given equation of the circle is:

\(x^2 + y^2 - 4x - 4y + 4 = 0\)

To find the radius of this circle, we can compare it with the standard general form of a circle's equation, which is:

\(x^2 + y^2 + 2gx + 2fy + c = 0\)

Comparing the Equations to Find Parameters

By comparing the coefficients of the given equation with the standard general form, we can identify the values of \(g\), \(f\), and \(c\).

• Coefficient of \(x\): In the given equation, it's \(-4\). In the standard form, it's \(2g\). So, \(2g = -4\), which means \(g = -2\).
• Coefficient of \(y\): In the given equation, it's \(-4\). In the standard form, it's \(2f\). So, \(2f = -4\), which means \(f = -2\).
• Constant term: In the given equation, it's \(+4\). In the standard form, it's \(c\). So, \(c = 4\).

Calculating the Radius of the Circle

The formula for the radius (\(r\)) of a circle from its general equation is given by:

\(r = \sqrt{g^2 + f^2 - c}\)

Now, we substitute the values of \(g\), \(f\), and \(c\) that we found:

• Substitute \(g = -2\): \(g^2 = (-2)^2 = 4\)
• Substitute \(f = -2\): \(f^2 = (-2)^2 = 4\)
• Substitute \(c = 4\)

Plugging these values into the radius formula:

\(r = \sqrt{(-2)^2 + (-2)^2 - 4}\)

\(r = \sqrt{4 + 4 - 4}\)

\(r = \sqrt{8 - 4}\)

\(r = \sqrt{4}\)

\(r = 2\)

Therefore, the radius of the circle represented by the equation \(x^2 + y^2 - 4x - 4y + 4 = 0\) is 2 units.

Verifying the Radius Calculation

The calculated radius of a circle is 2. This value matches one of the options provided.