留学考试中数学的答题格式（用英语答题！！！）

Hereisanexample:

CallthetworaysR(containsP)andS(containsA,B)andlettheanglebetweenthembeα.

ItriedthisinCartesianform,butgeometryisprobablybetter.

WLOG,chooseoriginOandRasthe+x-axis.

ThepointsareA(a,0)B(b,0)

Let'sscaleby1/asuchthatA'ismappedto(1,0)andB'ismappedto(b/a,0).Let'scallthat(k,0).kissomeconstant.)

LetPbedistanceralongR(strictly,thatdistancehasbeenscaledby1/atoo).

Wearetryingtofindr(risgoingtodependonαandk)

CoordsofAPBarethusA(1,0)P(rcosα,rsinα)B(k,0)

LetangleAPB=θ,andapplytheCosineRule.

Sidelengthsare

|PA|=√((rcosα-k)²+(rsinα)²)=√(r²+k²-2krcosα)

|PB|=√((rcosα-1)²+(rsinα)²)=√(r²+1-2rcosα)

|AB|=(k-1)

Youcangeta(nasty)expressionforcosθintermsofparameterrandconstantsk,α.

Thenfindtherwhichmaximizesθ,byminimizingcosθ(regardlesswhether0<θ<π/2orπ/2<θ<π(cosinegoes-ve)).

Thiswillgettherightanswer,it'lljustbeawkward.

Achangeofnotationmightmakeitclearer.

Anyway,forwhatit'sworth:

cosθ=|PA|²+|PB|²-|AB|²/2|PA||PB|

=((r²+k²-2krcosα)+(r²+1-2rcosα)-(k-1)²)/(r²+k²-2krcosα)*(r²+1-2rcosα)

andsoon