HMMT 二月 2014 · 几何 · 第 5 题

英文解析

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 spacewith coordinates (3 , 4 , 8). Find the largest possible radius of a sphere that is contained entirely in theslanted 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 circleinscribed in a triangle with a vertex P . By symmetry, this circle is a great circle of the sphere, andhence 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
   °