To derive the equation of a circle in standard form given the center and the radius, we use the standard form equation of a circle:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
From the information provided:
- The center of the circle is at \((-8, -2)\), so \(h = -8\) and \(k = -2\).
- The radius \(r = 2\).
Now calculating \(r^2\): \[ r^2 = 2^2 = 4 \]
Substituting \(h\), \(k\), and \(r^2\) into the standard form equation:
\[ (x - (-8))^2 + (y - (-2))^2 = 4 \] This simplifies to: \[ (x + 8)^2 + (y + 2)^2 = 4 \]
Thus, the equation of the circle in standard form is:
\[ (x + 8)^2 + (y + 2)^2 = 4 \]
Now, looking at the choices:
- a. \((x + 2)^2 + (y + 8)^2 = 2\)
- b. \((x - 2)^2 + (y - 8)^2 = 2\)
- c. \((x - 8)^2 + (y - 2)^2 = 4\)
- d. \((x + 8)^2 + (y + 2)^2 = 4\)
The correct answer is: d. \((x + 8)^2 + (y + 2)^2 = 4\)