What is the equation for the translation of x^2 + y^2 = 25 two units to the left and four units down?
The equation for the translation of x^2 + y^2 = 25 two units to the left and four units down can be found by substituting (x + 2) for x and (y - 4) for y in the original equation:
(x + 2)^2 + (y - 4)^2 = 25
Expanding:
x^2 + 4x + 4 + y^2 - 8y + 16 = 25
Rearranging terms:
x^2 + y^2 + 4x - 8y + 20 = 25
Simplifying:
x^2 + y^2 + 4x - 8y - 5 = 0
Therefore, the equation for the translation is x^2 + y^2 + 4x - 8y - 5 = 0.