Question
On a coordinate plane, a circle has a center at (1, negative 2) and a radius of 4 units.
Which equation represents a circle with the same radius as the circle shown but with a center at (-1, 1)?
(x – 1)2 + (y + 1)2 = 16
(x – 1)2 + (y + 1)2 = 4
(x + 1)2 + (y –1)2 = 4
(x + 1)2 + (y – 1)2 = 16
Which equation represents a circle with the same radius as the circle shown but with a center at (-1, 1)?
(x – 1)2 + (y + 1)2 = 16
(x – 1)2 + (y + 1)2 = 4
(x + 1)2 + (y –1)2 = 4
(x + 1)2 + (y – 1)2 = 16
Answers
GPT-4o mini
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the new center of the circle is \((-1, 1)\), and the radius remains 4. Thus, we have:
- \(h = -1\)
- \(k = 1\)
- \(r = 4 \implies r^2 = 16\)
Plugging these values into the circle equation gives:
\[
(x - (-1))^2 + (y - 1)^2 = 16
\]
This simplifies to:
\[
(x + 1)^2 + (y - 1)^2 = 16
\]
Thus, the equation that represents the circle with the same radius but with a center at \((-1, 1)\) is:
\((x + 1)^2 + (y - 1)^2 = 16\)
So the correct choice is:
**\((x + 1)^2 + (y - 1)^2 = 16\)**.
\[
(x - h)^2 + (y - k)^2 = r^2
\]
In this case, the new center of the circle is \((-1, 1)\), and the radius remains 4. Thus, we have:
- \(h = -1\)
- \(k = 1\)
- \(r = 4 \implies r^2 = 16\)
Plugging these values into the circle equation gives:
\[
(x - (-1))^2 + (y - 1)^2 = 16
\]
This simplifies to:
\[
(x + 1)^2 + (y - 1)^2 = 16
\]
Thus, the equation that represents the circle with the same radius but with a center at \((-1, 1)\) is:
\((x + 1)^2 + (y - 1)^2 = 16\)
So the correct choice is:
**\((x + 1)^2 + (y - 1)^2 = 16\)**.