Final answer:
The radius of the original circle is 6 and the center is (-5,7). After the translation, the radius of the translated circle remains 6 and the new center is (-4,5).
Step-by-step explanation:
To translate a circle one unit right and two units down, we add 1 to the x-coordinate and subtract 2 from the y-coordinate of the center of the original circle. Since the equation of the circle is x² + y² + 10x - 14y + 38 = 0, the original center is (-5,7). After the translation, the new center is (-4, 5).
The radius of the original circle can be found by completing the square of the x and y terms. x² + 10x and y² - 14y can be rewritten as (x + 5)² - 25 and (y - 7)² - 49, respectively. So, the equation becomes (x + 5)² - 25 + (y - 7)² - 49 + 38 = 0. Simplifying it further, we have (x + 5)² + (y - 7)² = 36. Therefore, the radius of the original circle is 6.
After the translation, the equation of the translated circle is (x + 4)² + (y - 5)² = 36. The radius of the translated circle remains 6, and the center is now (-4,5).