Circle area from two-point radius: In the xy-plane, points P(2, 0) and Q(5, 4) are given. If a circle has radius equal to the distance PQ, find its area (use π as pi).
Aptitude
Elementary Algebra
Difficulty: Easy
Choose an option
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A16 π
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B32 π
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C14 π
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D25 π
Answer
Correct Answer: 25 π
Explanation
Introduction / Context:The question combines distance formula in the coordinate plane with the area formula for a circle. First compute the segment length PQ; that length becomes the radius r. Then evaluate area = π * r^2.
Given Data / Assumptions:
- P = (2, 0)
- Q = (5, 4)
- Area of circle = π * r^2 with r = PQ
Concept / Approach:Use r = sqrt[(x2 − x1)^2 + (y2 − y1)^2]. Then square r to compute area. This avoids working with square roots at the end since r^2 is needed directly in the area formula.
Step-by-Step Solution:
Compute differences: Δx = 5 − 2 = 3; Δy = 4 − 0 = 4.Distance squared: r^2 = 3^2 + 4^2 = 9 + 16 = 25.Thus r = 5.Area = π * r^2 = π * 25 = 25 π.Verification / Alternative check:
The 3-4-5 right triangle is a standard Pythagorean triple; distance 5 is expected, making area 25 π immediate.Why Other Options Are Wrong:
- 16 π, 14 π, 32 π: These correspond to incorrect computations of r^2; only 25 π matches Δx^2 + Δy^2 = 25.
Common Pitfalls:
- Using r instead of r^2 in the area formula or mixing up Δx and Δy.
Final Answer:
25 π