解析几何双语卷(07-11级)

1、07级a卷i fill in the blanks. (5 x6=30')1. if 万=(3,-6,-1),/? = (1,4,一5) and 0 = (3,-42), the scalar projection ofa+h onto c is 丄 and the vector projection of a + h onto cis .2 the normal form of the equation of the plane 龙：2兀一y + z + ll = 0is the points a(2,-1j) and 0(000) are on thesame side of th

2、e plane 7c or not (yes or no).2x + 3y = 0,3. the angle which the line makes with the plane 3x-2y + 7z = 87x-3z = 0,is .4. the distance from the plane 3x + 4y-z-5 = 0 to the point (1,1,0)is .5. the equation of the surface generated by revolving the curve+ y lax 0,."about the x-axis is z = (6. th

3、e equation of the projection j- 9y2 + 6xy- 2xz + 24x - 9y + 3z - 63 = 0, 2x-3y + z - 9 = 0,ii ( 10') if the angle between the vectors afind the angle between two vectors a + bupon the xy- plane of the curveis .and b is , a =4 and b = 1.(1) 6and a-b. (2) write the area of theparallelogram determi

4、ned by a and b .iil (80 find the equation of the plane through the lineparallel to the line /2 determined by two points a(2,5，-3) and b(3,-2,2).(14 j (1) prove that the two linesy-2z-33 randare not coplanar. (2) write the equation of commonperpendicular. (3) compute the distance between the given tw

5、o lines.v (10 j find the equation of the cylinder whose generating lines are parallel to the vector v = (5,3,2) and directrix is the curve c: <z = 0.vi (10 j(1) find the asymptotes of the hyperbola6x +5xy-6y -39x + 26y-13 = 0. (2) write the transformation of coordinates with the two asymptotes fo

6、r the new coordinate axes.vii. (10j determine the type of the conic x2 -4xy + 4y2 +2x-2y-l =0 by using the invariants.v11l (8 j prove that the two principal axes of the central conic ax1 + 2lvcy + ay2 = dare - y - 0.07级b卷i fill in the blanks.(59 x 6)1. suppose <7=3, |z?| = 26 , axh = 72 , then a-

7、b=.2 the volume of the parallelepiped with the vertexes a(0,0,0), 3(3,4,-1),c(2,3,5),0(6,0-3) is .ce , 1,x-3 y-2 z . x + 2y-3 z + 53. the angle between the lines = = f= and = =厂1-1 v2 11 v2is .-.2x+y + z-5 = 0,4 the symmetric form of the equations of the line <2x+y-3z-l = 0, is 5. the equation of

8、 the surface of revolution obtained by revolving the curvex2 -6z = 0.<about the z-axis isi y = oon6. the center of the conic 2x +5xy+ 2)汀一 6x-3y+ 5 = 0 is ii (79) suppose oa=rob= r2,oc = r3 write the area of mbc and prove: ' » > * if 斤 x q + 厂2 x f + $ x “ = °, then the points a

9、, b , c are collinear.iii. (10) find the equation of the line through p(2丄3) and perpendicular andx +1 _ y -1 _ zintersect to the line 32-1iv.(10，)find the equation of the plane through the linex + 5j + z = 0x-z + 4 = 0and makesthe angle 45°with the plane : x - 4 j - 8z +12 = 0.v . (10j suppose

10、 the equations of the directrix are=1,1.thegenerating lines are perpendicular to the plane determined by the directrix. find the equation of the cylindrical surface.vi (16 j use the invariants to simplify the equation of the conic8x2 +4xy + 5y2 +8x-16y-16 = 0and write the standard equation. if take

11、the principle diameters as the new coordinates axes and then simplify this equation, write the transformations of coordinates.vii. (10，)for what values of 入"isx2 + 6xy + ay2 + 3x + “y - 4 = 0an equation of a conic with (1) a unique center, (2) no center, (3) a central line. vih. (7') show i

12、f a conicanx r2anxy + a22y -2ai3x + 2a23y += 0has the asymptotes, then the equations of the asymptotes are<d(x-x0,y-y0)= h(x-x0)2 +2a12(x-x0)(y-y0) + tz22(y-y0)2=0,where (x0,) is the center of the conic.09级a卷i fill in the blanks. (4,x7=28,)1. if a = (3,-5,8),z? = (-1,1, z), a+b = a-b. the compone

13、nt z =.2. given (axb) c = 2 , then (a + )x(b + c) (c + 5) =x + 3y+ 2z + l = 03. the angle that the line i :<makes with the plane62x-y-10z + 3 = 0e龙：4x _ 2y + z = 2 is .4. the equation of the projecting cylinder of the curve* 1045 on thex-2y+ z = 0xy - plane is 5. the projecting point of the given

14、 point p(2,3)on the line ,兀+ 7 y + 2 z + 2 .i:= is1236. which of the points(3,5,2), p2 (1,6,-3) are one the same side of the plane5x-2y-3z = 0 as the point p(l,-4,3) ? . (or p2)7. the equation of the tangent at the origin to the conic 5x2 +7xy+ y2 一兀+ 2y = 0is .ii (10') given the four points(6,-

15、3,6), (5,3,2), (4,5,-l), m4(3,-6,1), (1)find the direction cosines of the vector m3m4. (2) write the area of the triangle with vertices. (3) find the volume of parallelepiped with adjacentedges mill(10) given the two linesz=2r=iandl2: x = y = z ,(1) write theequation of common perpendicular. (2) com

16、pute the distance between the given two lines.iv. (80 find the equation of plane passing through the line l:x + y z = 0 x-y+z+l=0andvertical to the plane 龙：x+y + z = 2. and write the equation for projecting line of the given line / on the plane 7i。v (&) find the equation of the circular cone wit

17、h vertice at the point m°(2丄2), axis perpendicular to the plane 71: 2x + 2y-z + l = 0, and angle between the generating line and the given axis.vii. (io') what is the equation of the surface generated by revolving the line = = about the z-axis? and discuss the type of the revolution surface

18、sa 01according to the value of a. 5.vi (2t)(1) determine the canonical equation of the conic- 3xy + y +10x-10y+ 21 = 0 by using the invariants. (2) find the equations of the principal axes of the conic. write the transformation of coordinates with the principal axes of the conic for the new coordina

19、te axes, and draw the figure of the conic. (3) describe the equations of the asymptotes if the conic is the hyperbola.viii(5)write the normal form of the equation of the plane = 1, and a b cprove that the distance from the plane 兀 to the origin o is + + = cr tr cr p_2010级a卷i fill in the blanks. (4 x

20、 10二40')1. the vector that has the same direction as a = (-2,4,2) but has length 6is2. given (axb)-c = 2 , then (方+ x)x(n +方)(e + n)=3. the normal form of the plane 龙：2x + 3y + 6z-49 = 0is , the distance to the origin from the plane 兀is.4. the projecting point of the point (-1,2,0) upon the plan

21、eyr : x + 2y- z + 1 = 0 is .5.6.the angle which the lineisz 3-makes with the plane x + y + z = 11-41x-1y-2the equation of the surface of revolution obtained by revolving the curveabout the y -axis is7.the equation of the projecting cylinder of the curve <z = &+2, on thex2 + j2 + z2 = 1,8.9.xy

22、 - plane is .which of the points 片(0丄 1)/(5丄 0) are on the same side of the planex-2y-z = 1 as the point p(l,l,0) ? . (or p2)x y z 2the distance from the origin to the line =1 2 1is10. the asymptotes of the hyperbola 6x2 - xy- y2 + 3x+ y- =0 are and x+y-z-l = 0,x-y + z + l = 0,11. (8') if a = 2,

23、b =5,z(a,b) = ,p = 3a-b,q = aa + nb, (1) write the scalar projection of a + b on a. (2) show the value of the coefficient a such that p 丄刁. (3)compute the area of the parallelogram with adjacent edges a and h .iii. (80 determine the equation of the plane through the line perpendicular to the plane 2

24、x+ y + z = l.iv. (10，)write the equation of the line through the point m (-1q4) and parallel toz the plane 龙：3x-4y + z-10 = 0 and also intersecting the line i ：x+ = y-3 = .f +），+ z? = 1,v - （10，）suppose the equations of the directrix are <，the兀+y + z = 0.generating lines are parallel to the line

25、l:二.find the equation of thecylindrical surface.vi.（8j find the equation of the tangent at the point m（2）to the conicx2 -xy+ y2 +2x-4y-3 = 0.v1l （16，）（1） determine the canonical equation of the conic x2 一 ay +-1 = 0 by using the invariants. （2） write the transformations of coordinates when referred

26、to the coordinate systems with the two principal axes.2011级a卷i . fill in the blanks. （49 x 10=40'）8. if 厅丄a =3 and | 方 |=4, |（& +方）x（& 方）|= a = 3, |z?| = 26 , axb = 72 , then9.a-b.the scalar projection ofb onto a is .10. the normal form ofis , theis.the plane龙：2x + 3y + 6z 35 = 0distance

27、 to the origin from the plane 7tx y 1 z 111. the point of intersection and the angle that the line z : = j =makeswith the plane /r： 2兀 + y z 3 = 0 are andrespectively.12. the projection of the curve c：x2 + y2 + z2 =9,.on the xy- plane x + z = is 13. the projecting point of the given point p（2,3j） on

28、 the line.x + 7 y + 2 z + 2 ./:= -= is.12314. which of the points 片（0丄 0）, £（6丄 0） are on the same side of the planex-2y-z = 1 as the point p（l,l,0）? .（片 or p2）& the equation of the surface obtained by revolving the curvex = fiy2about thez = 0x-axis is ca9. the center of the conic x -xyy +2x-4y-3 = 0 is10. the asymptotes of the hyperbola x2 xy-2y2 -2x + 5y-2 = 0are and ac but has length6. (2) write the area of the triangle11. (