2024 AMC 12B Problems/Problem 13

Solution 1: Easy

Adding up the first and second statement, we get: h+k = 2x^2 + 2y^2 - 16x - 4y

   = 2(x^2 - 8x) + 2(y^2 - 2y)
   = 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1)
   = 2(x - 4)^2 + 2(y - 1)^2 - 34

All squared values must be greater or equal to 0. As we are aiming for the minimum value, we let the 2 squared terms be 0. This leads to (h+k)min = 0 + 0 - 34 = (C) -34

~mitsuihisashi14