Which of the following is a result of shifting a circle with the equation
[tex](x-1)^2+(y-4)^2=16[/tex]
to the right 4 units?

A. The [tex]x[/tex]-coordinate of the center of the circle increases by 4.
B. Both the [tex]x[/tex]- and [tex]y[/tex]-coordinates of the center of the circle decrease by 4.
C. The [tex]y[/tex]-coordinate of the center of the circle increases by 4.
D. Both the [tex]x[/tex]- and [tex]y[/tex]-coordinates of the center of the circle increase by 4.



Answer :

Let's carefully analyze the problem step-by-step:

1. Understanding the initial equation of the circle: The given equation of the circle is [tex]\((x-1)^2 + (y-4)^2 = 16\)[/tex].

- This equation represents a circle where the center is at [tex]\((1, 4)\)[/tex] and the radius is 4 (since [tex]\( \sqrt{16} = 4 \)[/tex]).

2. Shifting the circle to the right by 4 units:

- When a geometric figure is shifted to the right by [tex]\(k\)[/tex] units, the [tex]\(x\)[/tex]-coordinates of all points on the figure increase by [tex]\(k\)[/tex] units, while the [tex]\(y\)[/tex]-coordinates remain unchanged.
- Therefore, if we shift our circle to the right by 4 units, the [tex]\(x\)[/tex]-coordinate of the center will increase by 4 units, and the [tex]\(y\)[/tex]-coordinate will remain the same.

3. Calculating the new center:
- The original center of the circle is at [tex]\((1, 4)\)[/tex].
- After shifting to the right by 4 units, the new [tex]\(x\)[/tex]-coordinate will be [tex]\(1 + 4 = 5\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains unchanged, so it will stay as 4.

Thus, the new center of the circle after the shift is [tex]\((5, 4)\)[/tex].

4. Identifying the correct option:
- Based on our new center [tex]\((5, 4)\)[/tex], we can see that:
- The [tex]\(x\)[/tex]-coordinate of the center has increased by 4 (from 1 to 5).
- The [tex]\(y\)[/tex]-coordinate of the center remains the same.

Hence, the correct answer is:

A. The [tex]\(x\)[/tex]-coordinate of the center of the circle increases by 4.