Question

use the graph of the circle to answer the question.
what is the equation of the circle written in the form ((x - h)^2 + (y - k)^2 = r^2)? enter the values of (h), (k), and (r) in the boxes.
(h = square)
(k = square)
(r = square)

Explanation:

Step1: Find the center \((h, k)\)

The center of the circle is the midpoint of the vertical and horizontal extremes. From the graph, the topmost point is \((2, 8)\) and the bottommost point is \((2, 0)\), the leftmost and rightmost can also be observed. The center \((h, k)\) is at \((2, 4)\) (since the vertical distance from top to bottom is \(8 - 0 = 8\), so radius is \(4\), and center is halfway, so \(y\)-coordinate is \(4\), \(x\)-coordinate is \(2\) as the horizontal center is at \(x = 2\)). So \(h = 2\), \(k = 4\).

Step2: Find the radius \(r\)

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

Answer:

\(h = 2\)
\(k = 4\)
\(r = 4\)