HMMT 二月 2024 · 冲刺赛 · 第 24 题

HMMT February 2024 — Guts Round — Problem 24

[12] A circle is tangent to both branches of the hyperbola x − 20 y = 24 as well as the x -axis. Compute the area of this circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2024, February 17, 2024 — GUTS ROUND Organization Team Team ID# ◦

解析

[12] A circle is tangent to both branches of the hyperbola x − 20 y = 24 as well as the x -axis. Compute the area of this circle. Proposed by: Karthik Venkata Vedula Answer: 504 π Solution 1: y ω x Invert about the unit circle centered at the origin. ω turns into a horizontal line, and the hyperbola turns into the following: 2 2 x 20 y 2 2 2 2 2 − = 24 = ⇒ x − 20 y = 24( x + y ) . 2 2 2 2 2 2 ( x + y ) ( x + y ) 4 2 2 4 2 = ⇒ 24 x + (48 y − 1) x + 24 y + 20 y = 0 2 2 4 2 = ⇒ (48 y − 1) ≥ 4(24)(24 y + 20 y ) 2 2 = ⇒ 1 − 96 y ≥ 1920 y √ = ⇒ y ≤ 1 / 2016 . √ This means that the horizontal line in question is y = 1 / 2016 . This means that the diameter of the √ circle is the reciprocal of the distance between the point and line, which is 2016 , so the radius is √ 504 , and the answer is 504 π . Solution 2: Let a be the y -coordinate of both tangency points to the hyperbola. Then, the equation of the circle must be in the form 2 2 2 x − 20 y + c ( y − a ) = 24 . 2 Comparing the y -coefficient, we see that c = 21 . Moreover, we need it to pass through (0 , 0) , so 2 21 a = 24 . Thus, the equation of the circle is 2 2 2 2 2 2 x + y − 42 ay + 21 a = 24 = ⇒ x + ( y − 21 a ) = (21 a ) , 2 so the radius is 21 a , and the area is (441 a ) π = 504 π . ◦