To determine which equation represents a circle that is concentric with the given circle but has a radius that is twice as large, let's follow these steps:
1. Identify the center and radius of the original circle:
The given equation is [tex]\((x - 4)^2 + (y - 6)^2 = 4\)[/tex].
This represents a circle centered at [tex]\((4, 6)\)[/tex] with a radius of [tex]\( \sqrt{4} = 2 \)[/tex].
2. Determine the center of the new circle:
Since the new circle is concentric with the given circle, it must have the same center. Therefore, the center of the new circle should also be [tex]\((4, 6)\)[/tex].
3. Calculate the radius of the new circle:
The radius of the new circle is twice as large as the radius of the original circle. So, the radius of the new circle is [tex]\(2 \times 2 = 4\)[/tex].
4. Write the equation of the new circle:
The general equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. For our new circle, [tex]\(h = 4\)[/tex], [tex]\(k = 6\)[/tex], and [tex]\(r = 4\)[/tex]:
[tex]\[
(x - 4)^2 + (y - 6)^2 = 4^2
\][/tex]
5. Simplify the equation:
[tex]\[
(x - 4)^2 + (y - 6)^2 = 16
\][/tex]
So, the equation that represents a circle which is concentric with the given circle but has a radius that is twice as large is:
[tex]\[
(x - 4)^2 + (y - 6)^2 = 16
\][/tex]
Thus, the correct choice is:
[tex]\((x-4)^2+(y-6)^2=16\)[/tex].
