HMMT 二月 2014 · 几何 · 第 5 题

HMMT February 2014 — Geometry — Problem 5

Let C be a circle in the xy plane with radius 1 and center (0 , 0 , 0), and let P be a point in space with coordinates (3 , 4 , 8). Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base C and vertex P .

解析

Let C be a circle in the xy plane with radius 1 and center (0 , 0 , 0), and let P be a point in space with coordinates (3 , 4 , 8). Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base C and vertex P . √ Answer: 3 − 5 Consider the plane passing through P that is perpendicular to the plane of the circle. The intersection of the plane with the cone and sphere is a cross section consisting of a circle inscribed in a triangle with a vertex P . By symmetry, this circle is a great circle of the sphere, and hence has the same radius. The other two vertices of the triangle are the points of intersection between 3 4 3 4 the plane and the unit circle, so the other two vertices are ( , , 0) , ( − , − , 0). 5 5 5 5 Using the formula A = rs and using the distance formula to find the side lengths, we find that √ 2 A 2 ∗ 8 √ r = = = 3 − 5. 2 s 2+10+4 5 ◦