A circle with the equation (x + 4)2 + (y - 3)2 = 9 is reflected over the line y = -1. What is the equation of the image?
Centre of the original circle (-4,3)
(-4,3) reflected over y=-1 is (-4, -5)
Note; 3-(-1) = (-1)-(-5) = 4
So the image is
(x+4)²+(y+5)²=9
reflections
To find the equation of the image when a circle is reflected over a line, we need to follow these steps:
Step 1: Determine the reflected coordinates of the center of the circle.
Step 2: Use the distance formula to find the radius of the reflected circle.
Step 3: Write the equation of the reflected circle.
Now, let's work through these steps to find the equation of the reflected circle in this specific problem:
Step 1: Determine the reflected coordinates of the center of the circle.
The line of reflection is y = -1. The center of the circle is given as (-4, 3). To find the reflected coordinates, we need to reflect the center point across the line y = -1.
When a point is reflected across a horizontal line, the y-coordinate remains the same, but the x-coordinate changes to its opposite.
So, the reflected coordinates of the center of the circle would be (-(-4), 3) = (4, 3).
Step 2: Use the distance formula to find the radius of the reflected circle.
The equation of the original circle is (x + 4)^2 + (y - 3)^2 = 9. We know that the radius is the square root of 9, which is 3.
Step 3: Write the equation of the reflected circle.
Using the reflected center (4, 3) and the same radius of 3, we can write the equation of the reflected circle as (x - 4)^2 + (y - 3)^2 = 9.
Therefore, the equation of the reflected circle is (x - 4)^2 + (y - 3)^2 = 9.