A circle with the equation (x + 3)2 + (y - 2)2 = 25 is reflected over the line x = 2.
What is the equation of the image?
nice lol
To find the equation of the image, we will first reflect the given circle over the line x = 2.
Step 1: Find the coordinates of the center of the given circle.
The equation of the given circle is (x + 3)^2 + (y - 2)^2 = 25.
Comparing this equation with the standard form of a circle equation, we can see that the center of the circle is (-3, 2).
Step 2: Find the equation of the image after reflecting over the line x = 2.
When a figure is reflected over a vertical line, such as x = 2, the x-coordinates of all points remain the same, while the y-coordinates change signs.
Since the x-coordinate of the center remains the same (-3), we only need to change the sign of the y-coordinate (2) to obtain the new center of the reflected circle, which is (-3, -2).
Therefore, the equation of the image circle after reflecting over the line x = 2 is given by:
(x - (-3))^2 + (y - (-2))^2 = 25
Simplifying this equation gives:
(x + 3)^2 + (y + 2)^2 = 25
So, the equation of the image is (x + 3)^2 + (y + 2)^2 = 25.
To find the equation of the image, we will apply the reflection transformation. When a point is reflected over the line x = 2, the x-coordinate remains the same, but the y-coordinate changes sign.
First, let's find the center of the circle. The general equation of a circle can be written as (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center point, and r is the radius.
Comparing the given equation (x + 3)^2 + (y - 2)^2 = 25 with the general equation, we can see that the center is (-3, 2), and the radius is √25 = 5.
Now, let's reflect the center point (-3, 2) over the line x = 2.
Since the reflection of a point (a, b) over the line x = c is (2c - a, b), we can find the new center point as follows:
x' = 2c - x = 2(2) - (-3) = 4 + 3 = 7
y' = y = 2
Therefore, the new center point is (7, 2).
The radius remains the same, so it is still 5.
Now, we can write the equation of the reflected circle using the new center point and radius:
(x - 7)^2 + (y - 2)^2 = 5^2
Simplifying, we get:
(x - 7)^2 + (y - 2)^2 = 25
Therefore, the equation of the image is (x - 7)^2 + (y - 2)^2 = 25.
y does not change
x new = (2-x) + 2 = 4-x
x = 4-xnew
(4-xnew + 3)^2 - (y-2)^2 = 25
(-xnew +7)^2 - (y-2)^2 = 25
same as
(xnew-7)^2 - (y-2)^2 = 25
check
old center was at x = -3
that is 5 to the left of x = 2 line
5 to the right of the line is x = 7
sure enough