SAT Circles Practice Question 10 (Hard)

Math · Geometry and TrigonometryHard

A circle in the $xy$ -plane has equation $x^2 + y^2 - 2x + 4y - 20 = 0$ . Which of the following points lies in the interior of the circle?

$(4, 2)$

$(5, -6)$

$(-3, 1)$

$(-2, 1)$

Show answer and explanation

Answer: D

Complete the square: $(x^2 - 2x + 1) + (y^2 + 4y + 4) = 20 + 1 + 4$ , so $(x - 1)^2 + (y + 2)^2 = 25$ . The center is $(1, -2)$ and the radius is $5$ . A point is inside if its squared distance from the center is less than $25$ . For $(-2, 1)$ : $(-3)^2 + 3^2 = 18 < 25$ , so it is inside. The points $(4, 2)$ and $(-3, 1)$ give exactly $25$ (on the circle), and $(5, -6)$ gives $32$ (outside).

Choices A and C lie exactly on the circle, not in its interior; their squared distance from the center equals $25$ .

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