2024 AMC 12B Problem 13

There are real numbers

x,

y,

h,

and

k

that satisfy the system of equations

x^2 + y^2 - 6x - 8y = h

x^2 + y^2 - 10x + 4y = k.

What is the minimum possible value of

h + k?

Show solution

Solution

Adding the equations,

\begin{aligned} h + k &= 2x^2 + 2y^2 - 16x - 4y \\ &= 2(x - 4)^2 \\ &\quad {}+ 2(y - 1)^2 - 34. \end{aligned}

Both squared terms are nonnegative, so the minimum occurs at

x = 4,

y = 1,

giving

h + k = -34.

Thus, the correct answer is C.

Tagged: completing the square · optimization