Question

two circles are concentric if they have the same center. which equation represents a circle that is concentric with the circle shown but has a radius that is twice as large?
$(x - 4)^2+(y - 6)^2 = 4$
$(x - 4)^2+(y - 6)^2 = 16$
$(x - 6)^2+(y - 4)^2 = 16$
$(x - 6)^2+(y - 4)^2 = 4$

Explanation:

Step1: Find the center and radius of the given circle

From the graph, the center of the given circle is \((4,6)\). By counting the grid - squares, the radius \(r = 2\) (since the distance from the center to the circle along the grid). The general equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Determine the new radius

The new radius is twice the original radius. Given \(r_{original}=2\), then \(r_{new}=2\times2 = 4\)

Step3: Write the equation of the new circle

Since the new circle is concentric with the given circle, its center \((h,k)=(4,6)\) and \(r = 4\). Substituting into the circle equation \((x - h)^{2}+(y - k)^{2}=r^{2}\), we get \((x - 4)^{2}+(y - 6)^{2}=4^{2}\)

Answer:

\((x - 4)^{2}+(y - 6)^{2}=16\) (the second option)