Circles SAT question

Math · Geometry and Trigonometry · Hard · Answer it here, explanation included.

A circle in the xy-plane has equation $open parenthesis, x plus 3, close parenthesis, squared, plus, open parenthesis, y minus 1, close parenthesis, squared, equals 25$ . Which of the following points does NOT lie in the interior of the circle?

ExplanationShow

Choice D is correct. The circle with equation $open parenthesis, x plus 3, close parenthesis, squared, plus, open parenthesis, y minus 1, close parenthesis, squared, equals 25$ has center $with coordinates negative 3 comma 1$ and radius 5. For a point to be inside of the circle, the distance from that point to the center must be less than the radius, 5. The distance between $the point with coordinates 3 comma 2$ and $the point with coordinates negative 3 comma 1$ is $the square root of, open parenthesis, negative 3 minus 3, close parenthesis, squared, plus, open parenthesis, 1 minus 2, close parenthesis, squared, end root, equals, the square root of, open parenthesis, negative 6, close parenthesis, squared, plus, open parenthesis, negative 1, close parenthesis, squared, end root, which equals the square root of 37$ , which is greater than 5. Therefore, $the point with coordinates 3 comma 2$ does NOT lie in the interior of the circle.

Choice A is incorrect. The distance between $the point with coordinates negative 7 comma 3$ and $the point with coordinates negative 3 comma 1$ is $the square root of, open parenthesis, negative 7 plus 3, close parenthesis, squared, plus, open parenthesis, 3 minus 1, close parenthesis, squared, end root, equals, the square root of, open parenthesis, negative 4, close parenthesis, squared, plus, open parenthesis, 2, close parenthesis, squared, end root, which equals the square root of 20$ , which is less than 5, and therefore $the point with coordinates negative 7 comma 3$ lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect because the distance between $the point with coordinates 0 comma 0$ and $the point with coordinates negative 3 comma 1$ is $the square root of, open parenthesis, 0 plus 3, close parenthesis, squared, plus, open parenthesis, 0 minus 1, close parenthesis, squared, end root, equals, the square root of, open parenthesis, 3, close parenthesis, squared, plus open parenthesis, 1, close parenthesis, squared, end root, which equals the square root of 8$ , which is less than 5, and therefore $the point with coordinates 0 comma 0$ in the interior of the circle.

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