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Test
Math
Domain
Geometry and Trigonometry
Skill
Circles
Difficulty
Hard
ID: 2266984b
Modded SAT Question Bank by Abdullah Mallik

x squared, plus 20 x, plus y squared, plus 16 y, equals negative 20

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?

  1. negative 20 comma negative 16

  2. negative 10 comma negative 8

  3. 10 comma 8

  4. 20 comma 16

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Correct Answer: B
Rationale

Choice B is correct. The standard equation of a circle in the xy-plane is of the form open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared, where the ordered pair h comma k are the coordinates of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, x squared, plus 20 x, needs a 100 to complete the square, and the sum of the second two terms, y squared, plus 16 y, needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields open parenthesis, x squared, plus 20 x, plus 100, close parenthesis, plus, open parenthesis, y squared, plus 16 y, plus 64, close parenthesis, equals negative 20, plus 100, plus 64, which is equivalent to open parenthesis, x plus 10, close parenthesis, squared, plus, open parenthesis, y plus 8, close parenthesis, squared, equals 144. Therefore, the coordinates of the center of the circle are negative 10 comma negative 8.

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.

 

Question Difficulty: Hard
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