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 \boxed{}. (type an equation. simplify your answer.)

Explanation:

Step1: Identify the center of the circle

The center of the circle is at the origin \((0,0)\) as it is the midpoint between \((-4,0)\) and \((4,0)\), and also between \((0,4)\) and \((0,-4)\). So, \(h = 0\) and \(k = 0\).

Step2: Determine the radius of the circle

The radius \(r\) is the distance from the center \((0,0)\) to a point on the circle, say \((4,0)\). Using the distance formula \(r=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(r=\sqrt{(4 - 0)^2+(0 - 0)^2}=\sqrt{16}=4\).

Step3: Write the standard equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = 0\), \(k = 0\), and \(r = 4\) into the formula, we get \((x - 0)^2+(y - 0)^2=4^2\), which simplifies to \(x^2 + y^2 = 16\).

Answer:

\(x^2 + y^2 = 16\)