Problem 14 - Entrance Test

If a circle with equation (x + 4)^2 + (y - 5)^2 = 16 is translated 3 units to the right and 2 units down, what is the new equation of the circle?

A. A) (x + 1)^2 + (y - 3)^2 = 16B. B) (x + 7)^2 + (y - 7)^2 = 16C. C) (x - 1)^2 + (y - 3)^2 = 16D. D) (x - 7)^2 + (y - 7)^2 = 16

Correct: A

The original circle has its center at (h, k) = (-4, 5) and radius r = sqrt(16) = 4. When a circle is translated: - 3 units to the right means adding 3 to the x-coordinate of the center: New x-coordinate = -4 + 3 = -1. - 2 units down means subtracting 2 from the y-coordinate of the center: New y-coordinate = 5 - 2 = 3. The new center is (-1, 3). The radius remains unchanged during a translation. So, the new equation of the circle is (x - (-1))^2 + (y - 3)^2 = 16, which simplifies to (x + 1)^2 + (y - 3)^2 = 16.