Question

write the standard equation for each of the circles in parts (a) through (e). 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

From the graph, the center of the circle is at the origin \((0,0)\) because it is the midpoint between \((-8,0)\) and \((8,0)\), and also between \((0,8)\) and \((0,-8)\). So, \(h = 0\) and \(k = 0\) in the standard circle equation \((x - h)^2+(y - k)^2=r^2\).

Step2: Determine the radius of the circle

The radius \(r\) is the distance from the center \((0,0)\) to any point on the circle, such as \((8,0)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), the distance from \((0,0)\) to \((8,0)\) is \(\sqrt{(8 - 0)^2+(0 - 0)^2}=\sqrt{64}=8\). So, \(r = 8\).

Step3: Write the standard equation of the circle

Substitute \(h = 0\), \(k = 0\), and \(r = 8\) into the standard circle equation \((x - h)^2+(y - k)^2=r^2\). We get \((x - 0)^2+(y - 0)^2=8^2\), which simplifies to \(x^2 + y^2 = 64\).

Answer:

\(x^2 + y^2 = 64\)