Question 1

The magnitude of projection of line joining (3,4, 5) and (4,6,3) on the line joining (-1,2,4) and (1,0,5) is

Accepted Answer

We know that, projection of a on b is given by projection, | a | = ( a b ) | b | Let line joining points (3,4,5) and (4,6,3) is L_1 and line joining points (-1,2,4) and (1,0,5) is L_2 aligned & L_1= i +2 j -2 k \ & L_2=2 i -2 j + k aligned Projection of L_1 and L_2= L_1 L_2 |L_2 | = 2-4-2 4+4+1 = -4 3 Magnitude = 4 3

Question 2

The angle between the lines whose direction cosines are given by the equations 3 l+m+5 n= 0,6 ~nm -2 n l+5 l m =0 is

Accepted Answer

The given equations are 3 l+m+5 n=0 ...(i) and 6 mn -2 n l+5 l m =0 ...(ii) From (i), we have m=-3 l-5 n. Putting m =-3 l-5 n in (ii), we get 6(-3 l-5 n) n-2 n l+5 l(-3 l-5 n)=0 (n+l)(2 n+l)=0 either l=- n or l=-2 n . If l=-n, then putting l=-n in (i), we obtain m=-2 n. If l=-2 n , then putting l=-2 n in (i), we obtain m = n . Thus, the direction ratios of two lines are - n ,- 2 n , n and -2 n , n , n i.e., 1,2,-1 and -2,1,1. Hence, the direction cosines are 1 6 , 2 6 , -1 6 or -2 6 , 1 6 , 1 6

Question 3

Let the acute angle bisector of the two planes x-2 y-2 z+1=0 and 2 x-3 y-6 z+1=0 be the plane P. Then which of the following points lies on P?

Accepted Answer

Given that P_1: x-2 y-2 z+1=0 P_2: 2 x-3 y-6 z+1=0 Equation of plane bisectors aligned & | x-2 y-2 z+1 1+4+4 |= | 2 x-3 y-6 z+1 2^2+3^2+6^2 | \ & x-2 y-2 z+1 3 = 2 x-3 y-6 z+1 7 aligned Since a _1 a _2+ b _1 ~b _2+ c _1 c _2=20 0 Negative sign will be taken for acute bisector. aligned & 7 x-14 y-14 z+7=-[6 x-9 y-18 z+3] \ & 13 x-23 y-32 z+10=0 \ & (-2,0,- 1 2 ) satisfy it aligned

Question 4

Let the foot of perpendicular from a point P (1,2,-1) to the straight line L : x 1 = y 0 = z -1 be N. Let a line be drawn from P parallel to the plane x+y+2 z=0 which meets L at point Q. If is the acu

Accepted Answer

Let x 1 = y 0 = z -1 = N ( , 0,- ) b = i - k PN b =0 1(1- )-( -1)=0 =1 N (1,0,-1) Let Q ( , 0,- ) n = i + j +2 k Now, PQ n =0 =-1 Q (-1,0,1) PN =2 j and PQ =2 i +2 j -2 k = 1 3

Question 5

If the co-ordinates of the point in which the line joining the points (3,5,-7) and (-2,1,8) is intersected by the plane y z is [ a , 13 ~b , c ], then (a+b-c)=

Accepted Answer

Ratio =- ( 3 -2 )= 3 2 Required co-ordinates of the points are aligned & [ 6-6 5 , 10+3 5 , -14+24 5 ]= (0, 13 5 , 2 ) . \ & a+b-c=0+5-2=3 aligned

Question 6

If the straight lines x-1 k = y-2 2 = z-3 3 and x-2 3 = y-3 k = z-1 2 intersect at a point, then the integer k is equal to

Accepted Answer

When the two lines intersect then shortes distance between them is zero i.e. aligned & ( a _2- a _1 ) b _1 b _2 | b _1 b _2 | =0 \ & ( a _2- a _1 ) b _1 b _2=0 aligned where a _1= i +2 j +3 k , b _1=k i +2 j +3 k aligned & a _2=2 i +3 j + k , b _2=3 i +k j +2 k \ & | array ccc 1 & 1 & -2 \ k & 2 & 3 \ 3 & k & 2 array |=0 \ & 1(4-3 k)-1(2 k-9)-2 (k^2-6 )=0 \ & -2 k^2-5 k+25=0 k=-5 or 5 2 aligned

Question 7

The shortest distance between the lines x = y + 2=6 z-6 and x+1=2 y=-12 z is

Accepted Answer

Question 8

The ratio in which the YZ-plane divide the line segment formed by joining the points (-2,4,7) and (3,-5,8) is 2: m . The value of m is

Accepted Answer

Let the points be A(-2,4,7) and B(3,-5,8) on Y Z-plane, x-coordinate =0. Let the ratio be K: 1. The coordinates of C are ( 3 K-2 K+1 , -5 K+4 K+1 , 8 K+7 K+1 ) Clearly 3 K-2 K+1 =0 3 K=2 K= 2 3 Hence required ratio is 2: 3.

Question 9

The lines x-2 1 = y-3 1 = z-4 -k and x-1 k = y-4 2 = z-5 1 are coplanar if

Accepted Answer

Two planes are coplanar if | array ccc x_2-x_1 & y_2-y_1 & z_2-z_1 \ l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 array |=0 | array ccc 1 & -1 & -1 \ 1 & 1 & -k \ k & 2 & 1 array |=0 Applying C _2 C _2+ C _1, C _3 C _3+ C _1 | array ccc 1 & 0 & 0 \ 1 & 2 & 1-k \ k & k+2 & 1+k array |=0 1[2+2 k-(k+2)(1-k)]=0 2+2 k- (-k^2-k+2 )=0 k^2+3 k=0 k(k+3)=0

Question 10

The shortest distance between the lines x-3 2 = y-2 3 = z-1 -1 and x+3 2 = y-6 1 = z-5 3 is :

Accepted Answer

Give lines are x-3 2 = y-2 3 = z-1 -1 and x+3 2 = y-6 1 = z-5 3 Lines passes through the points a _1=(3,2,1) and a _2=(-3,6,5) , b _1=2 i +3 j - k b _1=2 i + j -3 k , a _2- a _1=6 i -4 j-4 k Shortest distance = | ( a _2- a _1 ) ( b _1 b _2 ) | | ( b _1 b _2 ) | b _1 b _2= | array ccc i & j & k \ 2 & 3 & -1 \ 2 & 1 & 3 array |=10 i -8 j -4 k ( a _2- a _1 ) ( b _1 b _2 )=60+32+16=108 | b _1 b _2 |= 100+64+16 = 180 S . D= 108 180 = 108 6 5 = 18 5

Question 11

If the plane 3 x+y+2 z+6=0 is parallel to the line 3 x-1 2 b =3-y= z-1 a , then the value of 3 a+3 b is

Accepted Answer

Given plane 3 x+y+2 z+6=0 and line x- 1 3 2 b 3 = y-3 -1 = z-1 a Since, plane is parallel to line, then aligned & 3 ( 2 b 3 )+(1)(-1)+2(a)=0 \ & 2 b-1+2 a=0 \ & a+b= 1 2 \ & aligned Now, 3 a+3 b= 3 2

Question 12

The length of the perpendicular drawn from the point (3,-1,11) to the line x 2 = y-2 3 = z-3 4 is :

Accepted Answer

Let feet of perpendicular is (2 , 3 +2,4 +3) Direction ratio of the line is 2 -3,3 +3,4 -8. and Direction ratio of the line are 2,3,4 2(2 -3)+3(3 +3)+4(4 -8)=0 =1 Feet of is (2,5,7) Length is 1^ 2 +6^ 2 +4^ 2 = 53

Question 13

A line makes angles of 45^ and 60^ with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is

Accepted Answer

Given =45^ , =60^ , =? array l ^ 2 + ^ 2 + ^ 2 =1 \ ^ 2 =1- 1 2 - 1 4 = 1 4 \ y =60^ or 120^ array

Three Dimensional Geometry — Previous Year Questions & Solutions