A circle with the equation (x+3)squared plus (y-2)squared = 25 is reflected over the line x=2. What is the equation of the image?
(x-3)^2 + (y+2)^2=25
(x-7)^2 + (y+2)^2=25
(x-7)^2 + (y-7)^2=25
(x-3)^2 + (y-2)^2=25
since the axis of reflection is vertical, the y-coordinates do not change.
(x,y) --> (2-(x-2),y) = (4-x,y)
The old center was (-3,2) so the new center is (7,2)
I don't see that as a choice. Typo in problem?
To find the equation of the image, we need to reflect the given circle over the line x=2.
Step 1: Translate the line x=2 to the origin by subtracting 2 from both sides of the equation.
x - 2 = 0
Step 2: Apply the reflection transformation.
Since the line x=2 is a vertical line, the reflection of a point (x, y) over x=2 can be found by calculating the difference between the x-coordinate of the original point and the x-coordinate of the reflection point, and then adding it to the x-coordinate of the reflection line (x=2).
For x-coordinate:
If the original point is (x, y), the reflection point is (2 - (x-2), y) = (4 - x, y).
For y-coordinate:
Since the line x=2 is a vertical line, the y-coordinate remains the same. So, the reflection point is (4 - x, y).
Step 3: Substitute the new x-coordinate (4 - x) and the y-coordinate (y) into the equation of the circle to find the equation of the image.
(x+3)^2 + (y-2)^2 = 25
Substitute x = 4 - x and leave y as it is:
(4 - x + 3)^2 + (y - 2)^2 = 25
Simplify the equation:
(x - 1)^2 + (y - 2)^2 = 25
Therefore, the equation of the image circle, after reflecting the given circle over the line x=2, is (x - 1)^2 + (y - 2)^2 = 25.
So, none of the provided options is the correct equation of the image. The correct answer is:
(x - 1)^2 + (y - 2)^2 = 25.