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.
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.
