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? $(x - 4)^{2}+(y - 6)^{2}=4$ $(x - 6)^{2}+(y - 4)^{2}=4$ $(x - 4)^{2}+(y - 6)^{2}=16$ $(x - 6)^{2}+(y - 4)^{2}=16$

Explanation:

Step1: Recall the standard form of a circle's 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: Determine the center and radius of the given circle

From the graph, the center of the given circle is \((4,6)\). For the equation \((x - 4)^2+(y - 6)^2 = 4\), comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(r^2 = 4\), so \(r = 2\).

Step3: Find the new radius

We want a radius that is twice as large. If the original radius \(r=2\), the new radius \(R = 2r=4\). Then \(R^2=16\).

Step4: Write the equation of the new circle

Since the new circle is concentric (same center \((h = 4,k = 6)\)) and \(R^2=16\), the equation is \((x - 4)^2+(y - 6)^2=16\).

Answer:

\((x - 4)^2+(y - 6)^2 = 16\) (the third option)