解析几何双语卷(07-11级)

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07级a卷

Ⅰ.Fillintheblanks.(5’×6=30’)

1.Ifand,thescalarprojectionofontois,andthevectorprojectionofontois.

2．Thenormalformoftheequationoftheplaneis.Thepointsandareonthesamesideoftheplaneornot(yesorno).

3．Theanglewhichthelinemakeswiththeplaneis.

4.Thedistancefromtheplanetothepointis.

5．Theequationofthesurfacegeneratedbyrevolvingthecurveabouttheaxisis.

6.Theequationoftheprojectionupontheplaneofthecurveis.

Ⅱ．(10’)Iftheanglebetweenthevectorsandis,and(1)Findtheanglebetweentwovectorsand.(2)Writetheareaoftheparallelogramdeterminedbyand.

Ⅲ．（8’）Findtheequationoftheplanethroughthelineparalleltothelinedeterminedbytwopointsand

Ⅳ．(14’)(1)Provethatthetwolinesandarenotcoplanar.(2)Writetheequationofcommonperpendicular.(3)Computethedistancebetweenthegiventwolines.

Ⅴ．(10’)Findtheequationofthecylinderwhosegeneratinglinesareparalleltothevectoranddirectrixisthecurve

Ⅵ．(10’)(1)Findtheasymptotesofthehyperbola(2)Writethetransformationofcoordinateswiththetwoasymptotesforthenewcoordinateaxes.

Ⅶ．(10’)Determinethetypeoftheconicbyusingtheinvariants.

Ⅷ．(8’)Provethatthetwoprincipalaxesofthecentralconicare

07级b卷

Ⅰ.Fillintheblanks.(5’×6)

1.Suppose，，，then.

2．Thevolumeoftheparallelepipedwiththevertexesis.

3．Theanglebetweenthelinesandis.

4．Thesymmetricformoftheequationsofthelineis.

5.Theequationofthesurfaceofrevolutionobtainedbyrevolvingthecurveaboutthez-axisis.

6.Thecenteroftheconicis.

Ⅱ．(7’)Suppose.Writetheareaof.Andprove:

If，thenthepointsA,B,Carecollinear.

Ⅲ．（10’）FindtheequationofthelinethroughP(2,1,3)andperpendicularandintersecttotheline.

Ⅳ．(10’)Findtheequationoftheplanethroughthelineandmakestheangle45°withtheplane.

Ⅴ．(10’)SupposetheequationsofthedirectrixareThegeneratinglinesareperpendiculartotheplanedeterminedbythedirectrix.Findtheequationofthecylindricalsurface.

Ⅵ．(16’)Usetheinvariantstosimplifytheequationoftheconic

andwritethestandardequation.Iftaketheprinciplediametersasthenewcoordinatesaxesandthensimplifythisequation,writethetransformationsofcoordinates.

Ⅶ．（10’）Forwhatvaluesofis

anequationofaconicwith(1)auniquecenter,(2)nocenter,(3)acentralline.

Ⅷ．（7’）Showifaconic

hastheasymptotes,thentheequationsoftheasymptotesare

whereisthecenteroftheconic.

09级a卷

Ⅰ.Fillintheblanks.(4’×7=28’)

Ifthecomponent＝.

Given，then.

Theanglethatthelinemakeswiththeplaneis.

Theequationoftheprojectingcylinderofthecurveontheplaneis.

Theprojectingpointofthegivenpointonthelineis.

Whichofthepointsareonethesamesideoftheplane

asthepoint?.(or)

Theequationofthetangentattheorigintotheconicis.

Ⅱ．(10’)Giventhefourpoints（1）Findthedirectioncosinesofthevector.（2）Writetheareaofthetrianglewithvertices.（3）Findthevolumeofparallelepipedwithadjacentedges.

Ⅲ．(10’)Giventhetwolinesand,(1)Writetheequationofcommonperpendicular.(2)Computethedistancebetweenthegiventwolines.

Ⅳ．（8’）Findtheequationofplanepassingthroughthelineandverticaltotheplane.Andwritetheequationforprojectinglineofthegivenlineontheplane。

Ⅴ．(8’)Findtheequationofthecircularconewithverticeatthepoint,axisperpendiculartotheplaneandanglebetweenthegeneratinglineandthegivenaxis.

Ⅶ．(10’)Whatistheequationofthesurfacegeneratedbyrevolvingthelineabouttheaxis?Anddiscussthetypeoftherevolutionsurfacesaccordingtothevalueof.

Ⅵ．(21’)(1)Determinethecanonicalequationoftheconicbyusingtheinvariants.(2)Findtheequationsoftheprincipalaxesoftheconic.Writethetransformationofcoordinateswiththeprincipalaxesoftheconicforthenewcoordinateaxes,anddrawthefigureoftheconic.(3)Describetheequationsoftheasymptotesiftheconicisthehyperbola.

Ⅷ．(5’)Writethenormalformoftheequationoftheplane,andprovethatthedistancefromtheplanetotheoriginis.

2010级a卷

Ⅰ.Fillintheblanks.(4’×10=40’)

Thevectorthathasthesamedirectionasbuthaslengthis.

Given，then.

Thenormalformoftheplaneis,thedistancetotheoriginfromtheplaneis.

Theprojectingpointofthepointupontheplaneis.

Theanglewhichthelinemakeswiththeplaneis.

Theequationofthesurfaceofrevolutionobtainedbyrevolvingthecurveaboutthe-axisis.

Theequationoftheprojectingcylinderofthecurveontheplaneis.

Whichofthepointsareonthesamesideoftheplane

asthepoint?.(or)

Thedistancefromtheorigintothelineis.

Theasymptotesofthehyperbolaare

and.

Ⅱ.(8’)If(1)Writethescalarprojectionofon.（2）Showthevalueofthecoefficientsuchthat.(3)Computetheareaoftheparallelogramwithadjacentedgesand.

Ⅲ.(8’)Determinetheequationoftheplanethroughthelineperpendiculartotheplane.

Ⅳ.(10’)Writetheequationofthelinethroughthepointandparalleltotheplaneandalsointersectingtheline.

Ⅴ．(10’)SupposetheequationsofthedirectrixareThegeneratinglinesareparalleltotheline.Findtheequationofthecylindricalsurface.

Ⅵ．(8’)Findtheequationofthetangentatthepointtotheconic

Ⅶ．(16’)(1)Determinethecanonicalequationoftheconicbyusingtheinvariants.(2)Writethetransformationsofcoordinateswhenreferredtothecoordinatesystemswiththetwoprincipalaxes.

2011级a卷

Ⅰ.Fillintheblanks.(4’×10=40’)

If.

，，，then,thescalarprojectionofontois.

Thenormalformoftheplaneis,thedistancetotheoriginfromtheplaneis.

Thepointofintersectionandtheanglethatthelinemakeswiththeplaneareandrespectively.

Theprojectionofthecurveontheplaneis.

Theprojectingpointofthegivenpointonthelineis.

Whichofthepointsareonthesamesideoftheplane

asthepoint?.(or)

8.Theequationofthesurfaceobtainedbyrevolvingthecurveaboutthe-axisis.

9.Thecenteroftheconicis.

10.Theasymptotesofthehyperbolaare

and