Question

write the standard equation for each of the circles in parts (a) through (c). the coordinates of the center and the radius for each circle are integers. (a) the equation of the circle in standard form is (type an equation. simplify your answer.)

Explanation:

Step1: Identify the center of the circle

From the graph, the center of the circle is at the origin? Wait, no, looking at the points: the center is at (0,0)? Wait, the points given are (-2,0), (2,0), (0,2), (0,-2). So the center (h,k) is (0,0) because it's the midpoint of these points.

Step2: Determine the radius

The radius \( r \) is the distance from the center (0,0) to any of the points, say (2,0). Using the distance formula \( r = \sqrt{(x - h)^2 + (y - k)^2} \), with (x,y)=(2,0) and (h,k)=(0,0), we get \( r = \sqrt{(2 - 0)^2 + (0 - 0)^2} = \sqrt{4} = 2 \).

Step3: Write the standard equation of a circle

The standard form of the equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where (h,k) is the center and \( r \) is the radius. Substituting \( h = 0 \), \( k = 0 \), and \( r = 2 \), we get \( (x - 0)^2 + (y - 0)^2 = 2^2 \), which simplifies to \( x^2 + y^2 = 4 \).

Answer:

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