Geometry and TrigonometryVery hardDiameter from a circle's general-form equation
Geometry and Trigonometry practice question
The equation x² + y² − 6x + 8y = 0 defines a circle in the xy-plane. What is the length of the circle's diameter?
Answer: 10
Complete the square: (x² − 6x + 9) + (y² + 8y + 16) = 9 + 16, so (x − 3)² + (y + 4)² = 25. The radius is √25 = 5, and the diameter is 2 × 5 = 10.
Common wrong answers
- If you answered 0
- 0 is the constant on the right of the original equation, but completing the square adds 9 and 16 to that side, giving r² = 25.
- If you answered 5
- 5 is the radius. The diameter is twice that.
- If you answered 25
- 25 is r², the right side after completing the square. Take the square root for r = 5, then double it.
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