HMMT 二月 2015 · 几何 · 第 8 题

英文解析

8. Let S be the set of discs D contained completely in the set { ( x, y ) : y < 0 } (the region below the
   2 3
   x -axis) and centered (at some point) on the curve y = x − . What is the area of the union of theelements of S ?4
   √
   2 π 3
   Answer: + Solution 1. An arbitrary point ( x , y ) is contained in S if and only if there
   0 0
   3 4
   2 3 2 2 2
   exists some ( x, y ) on the curve ( x, x − ) such that ( x − x ) + ( y − y ) < y , since the radius of the
   0 0
   2 2 34
   circle is at most the distance from ( x, y ) to the x -axis. Some manipulation yields x − 2 y ( x − ) −
   2 240
   2 xx + x + y < 0.
   0 00
   Observe that ( x , y ) ∈ S if and only if the optimal choice for x that minimizes the expression satisfies
   0 0
   0 xthe inequality. The minimum is achieved for x = . After substituting and simplifying, we obtain
   1 − 2 y − x 20
   2 2 3 2 2 3
   y ( + x + y + y ) < 0. Since y < 0 and 1 − 2 y > 0, we find that we need − 2 x − 2 y + − 2 y >0
   0 0 0 0 0
   0 0 0 0
   1 − 2 y 2 2
   0 ⇐⇒ 1 > x + ( y + ) .210
   0 0
   Geometry 2
   S is therefore the intersection of the lower half-plane and a circle centered at (0 , − ) of radius 1. This 1
   is a circle of sector angle 4 π/ 3 and an isosceles triangle with vertex angle 2 π/ 3. The sum of these areas 2
   √
   2 π 3
   is + .
   3 4
   Solution 2. Let O = (0 , − ) and ℓ = { y = − 1 } be the focus and directrix of the given parabola. Let 1
   ′ ′2
   ℓ denote the x -axis. Note that a point P is in S iff there exists a point P on the parabola in the lower
   ′ ′ ′
   half-plane for which d ( P, P ) < d ( P, ℓ ). However, for all such P , d ( P, ℓ ) = 1 − d ( P, ℓ ) = 1 − d ( P, O ),
   ′ ′
   which means that P is in S iff there exists a P on the parabola for which d ( P , P ) + d ( P, O ) < 1. Itis not hard to see that this is precisely the intersection of the unit circle centered at O and the lowerhalf-plane, so now we can proceed as in Solution 1.