PUMaC 2020 · 几何（B 组） · 第 6 题
PUMaC 2020 — Geometry (Division B) — Problem 6
题目详情
- Let C be a circle centered at point O, and let P be a point in the interior of C. Let Q be a point on the circumference of C such that P Q ⊥ OP, and let D be the circle with diameter P Q. Consider a circle tangent to C whose circumference passes through point P. Let the curve Γ be the locus of the centers of all such circles. If the area enclosed by Γ is 1 / 100 the area of C, then what is the ratio of the area of C to the area of D ?
解析
- Let C be a circle centered at point O, and let P be a point in the interior of C. Let Q be a point on the circumference of C such that P Q ⊥ OP, and let D be the circle with diameter P Q. Consider a circle tangent to C whose circumference passes through point P. Let the curve Γ be the locus of the centers of all such circles. If the area enclosed by Γ is 1 / 100 the area of C, then what is the ratio of the area of C to the area of D ? Proposed by: Ollie Thakar Answer: 2500 Let r be the radius of C, and let the length OP = x. First, we prove that Γ is an ellipse with foci at O and P. Let X be a point on Γ . Then, draw a circle E centered at X passing through point P, tangent to C. Since C and E are tangent circles, then O, X, and C are collinear. But XC = XP, so r = OC = OX + XC = OX + XP, so OX + XP is a constant for all X on the curve Γ , which is the definition of an ellipse. The area of Γ is equal to π times the semi-major axis times the semi-minor axis, or, after an √ r 1 2 2 application of the Pythagorean theorem: π · · r − x . 2 2 π 2 2 2 Also by the Pythagorean Theorem, QP = r − x , so that means the area of Γ is rQP. 4 2 π 2 By the condition that the area of C is 100 times that of Γ , then we get that πr = 100 rQP, 4 r 100 from which we conclude that = = 25 , but the ratio of the area of C to the area of D QP 4 2 r 2 is precisely the square of the ratio , which is (2 · 25) = 2500 . QP Note: We initially had the answer of 625 , but this is incorrect on account of QP being the diameter and not the radius of the circle. We apologize for the confusion this would have caused.