PUMaC 2013 · 几何（A 组） · 第 5 题

PUMaC 2013 — Geometry (Division A) — Problem 5

[ 5 ] Suppose you have a sphere tangent to xy -plane with its center having positive z -coordinate. 2 If it is projected from a point P = (0 , b, a ) to the xy -plane, it gives the conic section y = x . p If we write a = where p, q are integers, find p + q . q

解析

[ 5 ] Suppose you have a sphere tangent to xy -plane with its center having positive z -coordinate. 2 If it is projected from a point P = (0 , b, a ) to the xy -plane, it gives the conic section y = x . p If we write a = where p, q are integers, find p + q . q Solution If P is (strictly) above the sphere, the projected curve should become ellipse. If it is above, the projection should become hyperbola instead. Thus the height of the sphere (two 2 times the radius) should be exactly same as a , the height of P . Also as y = x is symmetric with respect to yz -plane, the sphere should also be symmetric. 1 2 Let Q = (0 , c, a/ 2) be the center of the sphere, and X = ( t, t , 0) be a point on the projected conic. Note that P X should be tangent to the sphere, so the distance from Q to P X is a/ 2. Using the inner product, this condition can be phrased as follows: ! ! a/ 2 P X · P Q = cos \ XP Q = . | P Q | | P X || P Q | ! ! 2 Meanwhile we have P X = ( t, t b, a ) and P Q = (0 , c b, a/ 2), so this becomes p a a ! ! 2 2 2 2 2 2 | P X | = t + ( t b ) + a = P X · P Q = ( c b )( t b ) + a / 2 . 2 2 Squaring both sides gives 2 2 4 4 a a a a 2 2 2 2 2 2 2 2 ( t b ) + t + = ( c b ) ( t b ) + a ( c b )( t b ) + , 4 4 4 4 which should be identity for all t . Comparing the coe cients gives | c b | = a/ 2 and c b = 1 / 4, so a should be 1/2 .