PSAT Math Multiple-Choice Question 83: Answer and Explanation

Question: 83

What is the center of a circle with equation x² + y² - 2x + 8y + 8 = 0?

• A. (-1, 4)
• B. (1, -4)
• C. (-2, 8)
• D. (2, -8)

Correct Answer: B

Explanation:

B

The question asks for the center of the circle with the provided circle equation. The equation of a circle in standard form is (x - h)² + (y - k)² = r², where (h, k) are the coordinates of the circle's center and r is the radius. Start by grouping terms with the same variable together to rewrite the equation as x² - 2x + y² + 8y = -8. In order to rewrite the given equation in standard form, you must complete the square. Take the coefficient of the first linear term, -2x, and divide the coefficient by 2 to get -1. Then, square this result to get 1. Add 1 to both sides of the equation to get (x² - 2x + 1) + y² + 8y = -8 + 1. Then, do the same with the coefficient of the other linear term, 8y. Divide 8 by 2, which is 4, and then square that, which is 16. Add 16 to both sides to get (x² - 2x + 1) + (y² + 8y + 16) = -8 + 1 + 16. Finally, factor the groups of terms in parentheses on the left side and do the arithmetic on the right side to get (x - 1)² + (y + 4)² = 9. The coordinates of the center of the circle are given by (h, k) in the standard form, so, for this circle, the center is located at (1, -4). The correct answer is (B).