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2024 AMC 12B Problem 13

Problem 13 of 25IntermediateAlgebra

There are real numbers x,x, y,y, h,h, and kk that satisfy the system of equations x2+y2−6x−8y=hx^2 + y^2 - 6x - 8y = h x2+y2−10x+4y=k.x^2 + y^2 - 10x + 4y = k. What is the minimum possible value of h+k?h + k?

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Solution

Adding the equations, h+k=2x2+2y2−16x−4y=2(x−4)2+2(y−1)2−34. \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,x = 4, y=1,y = 1, giving h+k=−34.h + k = -34. Thus, the correct answer is C.

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Concepts: completing the square · optimization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.