SAT Circles Practice Question 1 (Hard)

Math · Geometry and TrigonometryHard

A circle in the $xy$ -plane has equation $x^2 + y^2 - 10x + 6y - 15 = 0$ . What is the radius of the circle?

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Answer: 7

Complete the square for each variable: $x^2 - 10x + 25$ and $y^2 + 6y + 9$ . Adding $25$ and $9$ to both sides of $x^2 - 10x + y^2 + 6y = 15$ gives $(x-5)^2 + (y+3)^2 = 49$ . The radius is $\sqrt{49} = 7$ .

Answering $15$ or $\sqrt{15}$ comes from reading the constant term without completing the square.

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