Question

question determine the equation of the circle graphed below.

Explanation:

Step1: Recall the standard circle equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify the center from the graph

Looking at the graph, the center of the circle appears to be at \((h, k) = (3, 5)\) (by observing the midpoint of the circle's position on the grid).

Step3: Determine the radius

To find the radius, we can look at the distance from the center to a point on the circle. For example, the circle intersects the y - axis at some points. Let's check the vertical or horizontal distance. From the center \((3,5)\) to a point on the circle, say, if we move horizontally or vertically. Looking at the grid, the radius seems to be 4 (by counting the grid units from the center to the edge of the circle). Let's verify: if the center is \((3,5)\) and we move 4 units in the x or y direction, we reach the circle's edge.

Step4: Substitute \(h\), \(k\), and \(r\) into the standard equation

Substitute \(h = 3\), \(k = 5\), and \(r = 4\) into \((x - h)^2+(y - k)^2=r^2\). We get \((x - 3)^2+(y - 5)^2 = 4^2=16\).

Answer:

\((x - 3)^2+(y - 5)^2 = 16\)