3D Geometry

THREE DIMENSIONAL GEOMETRY

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UNIT 07. THREE DIMENSIONAL GEOMETRY QUESTIONS FROM CBSE BOARD PAPERS(From 1992- )

Straight line

01.Find the equation of the line passing through the point with position vector 2i-3j-5k and perpendicular to the plane r.(6i-3j-5k)+2 = 0 [ Ans: r= 2i-3j-5k + (6i-3j-5k)]. 02. Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) on the line joining to the points B(1,4,6) and C(5,4,4). [Ans: (3,4,5)] 03. A (0,6,-9), B(-3,-6,3) and C(7,4,-1) are three points. Find the equation of the line AB if d is the foot of the perpendicular drawn from the point C to the line AB, find the coordinates and point D. [ Ans: x/1 = (y-6)/ 4 = (z+9)/-4 , (-1,2,-5)] 04.Find the foot of the perpendicular from the point (0,2,7) on the line x+2 = y-1 = z-2 . [ Ans: (-3/2,-1/2, 4)] 1 2 3 05. Find the shortest distance between the lines r= 4i-j+iI+2j-3k) and r= i-j + (2i+4j-5k). [ Ans: 6/5 units]

06. Find the vector equation of a line passing through a point with position vector 2i-j+k and which is parallel to the line joining the points with position vectors i+4j+k and i+2j+2k. Also, find the Cartesian equivalent of this equation. [Ans: 2i-j+k +t(2i-2j+k); x-2 = y+1 = z-1 ] 2 -2 1 07.The Cartesian equation of a line is 6x-2 = 3y+1 = 2z-2. Find a) the direction ratios of the line b) cartesian and vector equations of the line parallel to this line and passing through the point (2,-1,-1). [Ans:1,2,3; r= 2i-j-k+(i+2j+3k); x-2 = y+1 = z+1 ] 1 2 3 08.Find the foot of the perpendicular from the point (0,2,3) on the line x+3 = y-1 = z+4 [Ans: (2,3,-1) ] 5 2 3 09.Find the shortest distance between the lines whose vector equations are r = (1-t)i + (t-2)j + (3-2t)k and r = (s+1)I + (2s-1)j+(2s+1)k [ Ans: 8/29 units] 10. Find whether or not the two lines given below intersect: r = (2 +1)I (+1)j + (+1)k and r = (3+2)I (5+5)j+(2-1)k. [Ans: Do not intersect]


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11. Find the shortest distance between the lines whose vector equations are r = (-1)i +(+1)j- (1+)k and r = (1-)i +(2-1)j+ (+2)k. [Ans:5/2 units] 12. Define the line of shortest distance between two skew lines. Find the shortest distance and vector equation of the line of shortest distance between the two lines given by r=2i-3k+ (2i-j) and r = 4i+3k+ (3i+j+k). [Ans:30 units; r= 2i+j-3k+ t(-i-2j+5k)]

13. Find the image of the point (1,6,3) in the line x = y-1 = z-2

1 2 3 [Ans: (1,0,7)] 14. Show that the lines x-5=y-7 = z+3 and x-8 = y-4 = z-5 intersect each

4 4 -5 7 7 3 other. 15.Show that the lines x-1 = y+1 = z-1 and x-2 = y-1 = z+1 do not intersect 3 2 5 4 3 -2 eachother.

16. The cartesian equations of a line are 3x+1=6y-2 = 1-z. Find the fixed point through which it passes its direction ratios and also its vector equations.[CBSE 04]

17. Find the equation of the line passing through the point P(-1, 3,-2) and perpendicular to the line x = y = z and x+2 = y-1 = z+1 [ CBSE 2005] 1 2 3 -3 2 5

18. The vector equations of two lines are:

r = i+2j+3k + (i-3j+2k) and r = 4i+5j+6k + (2i+3j+k) Find the shortest distance between the above lines. [ CBSE D06].

Plane[s] 01. Find the equation of the plane through the point (3,4,-1)and the which is parallel to the plane r.(2i-3j+5k)+7 =0 [Ans: r.(2i-3j+5k)+ 11 = 0 ] 02.Find the equation of the plane through the intersection of the planes x+y+z = 6 and 2x+3y+4z +5 = 0 and which is passes through the point (1,1,1). [Ans:20x+23y+26z-69=0]

03. Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes x+2y+3z-7 =0 and 2x-3y+4z =0 [ Ans:17x+2y-7z= 26] 04. Find the equation of the plane through the points (2,-3,1) and (5,2,-1) and perpendicular to the plane x-4y+5z+2 =0 . [Ans: x-y-z = 4]


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05. Find the angle between the planes 2x-3y+4z= 1 and -x+y = 4 [Ans: cos-1 [5/58]]

06.The position vector of the points A and B are 3i+j+2k and i-2j-4k respectively. Find the vector equation of the plane passing through the

point B and perpendicular to the vector AB. [Ans: r.(2i +3j+ 6k) + 28 = 0]. 07. Find the equation of the plane passing through the points (3,4,2) and (7,0,6) and which is perpendicular to the plane 2x-5y = 15. [Ans: 5x+2y-3z=17] 08. Find the equation of the plane passing through the points (1,-1,2) and (2,-2,2) and which is perpendicular to the plane 6x-2y+2z = 9. [ Ans: x+y-2z+4 = 0] 09.Find the equation of the plane through the point (1,1,1) and perpendicular to each of the following planes:x+2y+3z=7and 2x-3y+4z= 0. [Ans:17x+2y-7z = 12].

10. Find the cartesian as well as vector equations of the planes through the intersection of the planes r.(2i+6j)=-12 and r.(3i-j+4k) = 0 which are at a unit distance from the origin. [Ans:2x+y+2z+3 = 0, x-2y+2z-3 =0 ; r.(2i+j+2k)+3 =0 , r.(i-2j+2k)-3 =0] 11. Find the vector equation of the plane passing through the intersection of the planes r.(2i-7j+4k)=0 and r.(3i-5j+4k)+11 =0 and passing through the point (-2,1,3). [Ans:r.(15i-47j+28k)-7 = 0] 12. Find the image of the point (1,3,4) in the plane x-y+z = 5.[ Ans : (3,1,6) ]

13.Find the length and the foot of the perpendicular from the point (1,1,2) to the plane r.(2i-2j+4k)+5 =0. [Ans:13 6/12 units; (-1/12,25/12,-1/12) ]

14. Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to the planes 3x+2y-3z =1 and 5x- 4y+z = 5. [CBSE 04] 15. Find the equation of the plane passing through the points (0,-1,0),(1,1,1) and (3,3,0). [CBSE 04]

16.Find the equation of the plane passing through the points P(1, -1, 2) and Q(2, -2, 2) and perpendicular to the plane 6x 2y + 2z = 9. [ CBSE 2005]

17. Find the image of the point (1,2,3) in the plane x+2y+4z = 38.[CBSE 06]


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18. Find the equation of the plane passing through the points (0,-1,-1),

(4,5,1)and (3,9,4).[ CBSE D 06] 19. Find the equation of the plane through the intersection of the planes r.(2i + j+ 3k) = 7, r.( 2i+5j+3k) = 9 and the point (2,1,3). [ CBSE 07] 20. Find the equation of the plane through the intersection of the planes r.(2i + j+ 3k) = 7, r.( 2i+5j+3k) = 9 and the point (3,2,-1). [ CBSE 07]

Plane and Lines

01. Find the equation in Cartesian form of the plane passing through the point (3,-3,1)and normal to the line joining the points (3,4,-1) and (2,-1,5)

[ Ans:x+5y-6z+ 18 =0] 02. Find the equation of the plane through the points (2,2,-1) and (3,4,2) and parallel to the line whose direction ratios are 7,0,6. [ Ans: 12x+ 15y 14z = 68] 03. Show that the line r= 2i-3j+5k+(I-j+2k) lies in the plane r.(3i+j-k)+2 = 0. 04. Find the distance between the point with position vector i-5j-10k and the point of intersection of the line x-2 = y+1 = z-2 with the plane x-y+ z = 5. 3 4 12 [ Ans:13 units] 05. Find the equation of the plane passing through the intersection of the planes 2x+3y-z=-1 and x+y-2z +3 = 0 and perpendicular to the plane 3x- y-2z=4 = 0 . [Ans: 7x+13y+4z-9= 0] 06.Find the equation of the plane passing through the point (1,2,1) and perpendicular to the line joining the points(1,4,2) and (2,3,5).Find the perpendicular distance of the origin from this plane. [ Ans: x-y-3z-2 =0 ,2/11] 07.Find the equation of the plane passing through the intersection of the planes x-2y+z = 1 and 2x+y+z=8 and parallel to the line with direction ratios1,2,1. Find also the perpendicular distance of the point (1,1,1) from this plane. [ Ans: 9x-8y + 7z 21= 0, 13/194 units] 08. Find the point R, where the line joining to the points P(1,3,4) and Q(-3,5,2) cuts the plane r.(2i-j+k) +3 =0. Is |PR | = |QR| . [ Ans: R(-1.4.3), yes]


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09. Show that the plane whose vector equation is r.(i+2j-k) = 1 and the line whose vector equation is r= -i+j+k +(2i+j+4k) are parallel. Also find the distance between them. [ Ans: 1/6 units] 10.A variable plane passes through a fixed point (1,2,3). Show that the locus of the foot of the perpendicular drawn from the origin to this plane is the sphere given by the equation x+y+z - x 2y-3z = 0. 11.A plane passes through a fixed point (1,-2,3) and cuts the axis in A,B and C. Show that the locus of the center of the sphere, passing through the points O,A,B and C is given by 1/x 2/y + 3/z= 2. 12. Find the distance of the point with position vector i-5j-10k from the point of intersection of the line r=2i-j+2k+(3i+4j+12k)with the plane r.(i-j+k)= 5. [Ans:13 units] 13. Show that the line whose vector equation is r = i+j+(2i+j+4k) is parallel to the plane whose vector equation is r.(I+2j-k) = 3 and find the distance between them. Also, state whether the line lies in the plane. [Ans:0; the line lies in the plane] 14. Find the vector equation of a line passing through the point, whose position vector is (2i-3j-5k) and perpendicular to the plane r.(6i-3j+5k)+2=0. Also, find the point of intersection of this line and the plane. [Ans:r=2i-3j-5k+t(6i-3j+5k);(76/35,-108/35,-34/35)] 15.Find the distance of the point (2,3,4) from the plane 3x+2y+5 =0 measured parallel to the line x+3 = y-2 = z . [ Ans: 7 units] 3 6 2

16.Show that the line L, whose vector equations r= 2i-2j+3k +(i-j+4k) is parallel to the plane , whose vector equation is r.(i+5j+k) = 5 and find the distance between them. [ Ans:10/27 units] 64. Find the coordinates of the point where the line x+1 = y+2 = z+3 meets the plane x+y+4z = 6. [ CBSE 06] 2 3 4 66. Find the equation of the plane through the points (1,2,3) and (0,-1,0) and parallel to the line x-1 = y+2 = z . [ CBSE D 06] 2 3 -3

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