Math: Geometry and Trigonometry

SAT Question of the Day for October 22, 2026

The equation x2+y2−14x+6y+33=0x^2 + y^2 - 14x + 6y + 33 = 0 represents a circle in the xyxy-plane. What is the radius of this circle?
  1. A.33
  2. B.55Correct answer
  3. C.77
  4. D.2525

Explanation

Rewrite the equation by completing the square. Group the xx and yy terms: (x2−14x)+(y2+6y)=−33(x^2 - 14x) + (y^2 + 6y) = -33. Complete the square for xx: x2−14x+49=(x−7)2x^2 - 14x + 49 = (x - 7)^2. Complete the square for yy: y2+6y+9=(y+3)2y^2 + 6y + 9 = (y + 3)^2. Adding 4949 and 99 to both sides gives (x−7)2+(y+3)2=−33+49+9=25(x - 7)^2 + (y + 3)^2 = -33 + 49 + 9 = 25. Since the standard form is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, the radius is r=25=5r = \sqrt{25} = 5. The distractor 33 comes from using only the yy-coordinate of the center, confusing a component of the center with the radius. The distractor 77 comes from reading the xx-coordinate of the center as the radius. The distractor 2525 results from reporting r2r^2 instead of rr.

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