Question

which of the following equations will produce this graph?
a. $(x - 3)^{2}+(y - 7)^{2}=4$
b. $(x - 3)^{2}+(y - 7)^{2}=16$
c. $(x + 3)^{2}+(y + 7)^{2}=4$
d. $(x + 3)^{2}+(y + 7)^{2}=16$

Explanation:

Step1: Recall the standard form of a circle equation

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

Step2: Identify the center and radius from the graph

From the graph, the center of the circle is \((3,7)\). The distance from the center \((3,7)\) to a point on the circle (e.g., \((3, 3)\) or \((7,7)\)) gives the radius. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (or by counting units on the coordinate - plane), if the center is \((3,7)\) and a point on the circle is \((7,7)\), then \(r=\sqrt{(7 - 3)^2+(7 - 7)^2}=\sqrt{16}=4\). So \(r^{2}=16\).

Step3: Substitute \(h = 3\), \(k = 7\), and \(r^{2}=16\) into the standard form

Substituting into \((x - h)^2+(y - k)^2=r^2\), we get \((x - 3)^2+(y - 7)^2=16\).

Answer:

B. \((x - 3)^2+(y - 7)^2 = 16\)