Geometry and TrigonometryVery hardCircle from the endpoints of a diameter
Geometry and Trigonometry practice question
In the xy-plane, the points (−2, 1) and (4, 9) are the endpoints of a diameter of a circle. Which equation represents the circle?
- A. (x − 1)² + (y − 5)² = 25Correct
- B. (x − 1)² + (y − 5)² = 100
- C. (x + 1)² + (y + 5)² = 25
- D. (x − 1)² + (y − 5)² = 5
Answer: A. (x − 1)² + (y − 5)² = 25
The centre is the midpoint: ((−2 + 4)/2, (1 + 9)/2) = (1, 5). The diameter is √(6² + 8²) = 10, so the radius is 5 and r² = 25.
Why the other answers are wrong
- B. (x − 1)² + (y − 5)² = 100
- 100 is the DIAMETER squared. The equation uses the radius squared: 5² = 25.
- C. (x + 1)² + (y + 5)² = 25
- (x + 1)² + (y + 5)² has centre (−1, −5); the centre is (1, 5), so the signs inside are minus.
- D. (x − 1)² + (y − 5)² = 5
- 5 is the radius itself. The right side of the equation is r² = 25.
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