HMMT 十一月 2023 · 冲刺赛 · 第 23 题

HMMT November 2023 — Guts Round — Problem 23

[12] The points A = (4 , ) and B = ( − 5 , − ) lie on the hyperbola xy = 1. The circle with diameter AB 4 5 intersects this hyperbola again at points X and Y . Compute XY . 2 k

解析

[12] The points A = (4 , ) and B = ( − 5 , − ) lie on the hyperbola xy = 1. The circle with diameter 4 5 AB intersects this hyperbola again at points X and Y . Compute XY . Proposed by: Pitchayut Saengrungkongka q 401 Answer: 5 Solution: X 1 A = (4 , ) 4 1 B = ( − 5 , − ) 5 Y Let A = ( a, 1 /a ), B = ( b, 1 /b ), and X = ( x, 1 /x ). Since X lies on the circle with diameter AB , we ◦ have ∠ AXB = 90 . Thus, AX and BX are perpendicular, and so the product of their slopes must be − 1. We deduce: a − x b − x = − 1 = ⇒ ( ax )( bx ) = − 1 , 1 1 1 1 − − a x b x √ √ √ √ √ so x = ± − ab . Plugging in a = 4 and b = − 5 gives X = ( 20 , 1 / 20) and Y = ( − 20 , − 1 / 20), giving the answer. 2 k