SAT Circles Practice Question 5 (Hard)

Math · Geometry and TrigonometryHard

In the $xy$ -plane, the graph of $x^2 + y^2 + 8x - 4y = k$ is a circle with radius $9$ . What is the value of $k$ ?

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Answer: 61

Complete the square: $x^2 + 8x + 16 + y^2 - 4y + 4 = k + 16 + 4$ , so $(x + 4)^2 + (y - 2)^2 = k + 20$ . The radius is $9$ , so $k + 20 = 81$ and $k = 61$ .

Answering $81$ forgets that $16$ and $4$ were added to the left side when completing the square, so they must be added to $k$ too.

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