12 - THREE DIMENSIONAL GEOMETRY( Answers at the end of all questions ) Page 1 (1) If the angle θ between the line 2x - y + 5 3 (a) (b) - 3 5 (c) 3 4 (d) - 4 3 (2) If the plane 2ax - 3ay + 4az + 6 = 0 passes through the midpoint of the line joining 2 2 2 the centres of the spheres x + y + z + 6x - 8y - 2z = 13 2 2 2 and x + y + z - 10x + 4y - 2z = 8, then a equals (a) -1 (b) 1 (c) -2 (d) 2 [ AIEEE 2005 ] → ra ^ ^ ce .c ^ ^ ^ (3) The distance between the line → r 2 i - 2 j + 3 k + λ ( i - j + 4 k ) and the plane (a) 10 9 r .( ^ ^ ^ i + 5 j + k ) = 5 s (b) m xa 10 c 3 3 3 10 (d) 10 3 (4) The angle between the lines 2x = 3y = - z and 6x = - y = - 4z is ( a ) 0° b ) 90° ( c ) 45° ( d ) 30° [ AIEEE 2005 ] w w .e (5) The p ane x + 2y - z = 4 cuts the sphere x of radius (b) 1 (c) 2 (d) 2 2 + y + z - x + z - 2 = 0 in a circle 2 2 w (a) 3 ( 6 ) A line makes the same angle θ with each of the X- and Z- axis. If the angle β, which 2 2 2 it makes with the y-axis, is such that sin β = 3 sin θ, then cos θ equals (a) 2 3 (b) 1 5 (c) 3 5 (d) 2 5 [ AIEEE 2004 ] om ^ y -1 x +1 z -2 = = and the plane 1 2 2 1 λ x + 4 = 0 is such that sin θ = , then the value of λ is 3 [ AIEEE 2005 ] [ AIEEE 2005 ] [ AIEEE 2005 ] 2y . are co-planar.3 ( b ) k = 1 or . y = 1 + t. a.12 . 3a ). 3a. 1. a. z = 1 + λs and x = [ AIEEE 2004 ] ( 10 ) The intersection of the sphe es x + y + z + 7x . 3a. 3a ).c 2 A line with direction cosines proportional to 2. a ) ( d ) ( 2a.3 om = z -5 1 [ AIEEE 2004 ] [ AIEEE 2004 ] [ AIEEE 2003 ] are coplanar. 2a.y .THREE DIMENSIONAL GEOMETRY ( Answers at the end of all questions ) Page 2 (7) Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is (a) 3 2 (b) 5 2 (c) 7 2 (d) 9 2 [ AIEEE 2004 ] (8) ( a ) ( 3a. z = c’y + d’ will be perpendicular if and only if ( a ) aa’ + cc’ + 1 = 0 ( c ) aa’ + bb’ = 0 and ( b ) aa’ + cc’ = 0 ( d ) aa’ + bb’ + cc’ = 0 w ( 12 ) The lines x -2 1 = y -3 1 = z -4 -k and x -1 = k y -4 2 ( a ) k = 0 or . 3a ).3 . ( a.2y .z = 1 ( d ) 2x . a. y = .z = ( c ) x . 2 meets each o the nes x = y + a = z and x + a = 2y = 2z. ( a.y . z = cy + d and x = a’y + b’.1 ( c ) k = 0 or . a. 2a ) ( b ) ( 3a.3x + 3y + 4z = 8 is the same as the intersection of one of the spheres and the plane (a) x .λs. 2a.z = 13 and 2 2 2 x + y + z .y 2z = 1 .t. then λ equals If the straight lines x = 1 + s. 3a ). ( a. if [ AIEEE 2003 ] . ( 2a.z = 1 w w ( 11 ) The ines x = ay + b. The coordinates of each of the p ints of intersection are given by (9) xa m (a) -2 (b) -1 (c) - 1 2 ra (d) 0 2 2 t .e ( b ) x . 2 z = 2 . a ) ( c ) ( 3a. a ) ce . with parameters s and t respectively.1 ( d ) k = 3 or . .y + z = 5 and parallel to the y z x line = is = 2 3 -6 (a) 1 (b) 7 (c) 3 [ AIEEE 2002 ] w ( 18 ) The co-ordinates of the point in which the line joining the points ( .2. b’ c’ from the origin. 3 14 . 5 5 13 ( b ) 0.2y . 3 ) from the plane x . 2 ( a ) 0. .e ( b ) 26 xa ( c ) 39 w w ( 17 ) The dist nce of a point ( 1.12 . 3 . 1.THREE DIMENSIONAL GEOMETRY ( Answers at the end of all questions ) Page 3 ( 13 ) Two systems of rectangular axes have the same origin.6z = 155 is ( a ) 13 . 2 .2y .2.7 ) and [ AIEEE 2002 ] . If a plane cuts them at distances a. ( c ) 0.c 14 2 1 . then (a) (c) 1 a2 1 + 1 + 1 + 1 + 1 + 1 = 0 (b) (d) 1 a2 1 a 2 + b 2 + c 2 ( 14 ) The direction cosines of the normal to the plane x + 2y . 3 14 (d) (b) 1 14 1 14 .3z (a) (c) 1 14 1 14 . 5. c and a’. ce . b. 5 13 2 . 8 ) and intersected by the YZ-plane are 13 . 14 3 14 2 2 2 ra (d) 4 ( d ) 11 ( d ) 13 ( 15 ) The radius of a circle in which the sphe e x + y + z the plane x + 2y + 2z + 7 = 0 is (a) 1 (b) 2 (c) 3 + 2x . . 5 5 om a' 2 b 2 c 2 a' 2 b' 2 c' 2 1 1 1 1 1 + = 0 2 2 2 2 2 a b c a' b' c' 2 + 1 b2 1 - 1 c2 1 + 1 a' 2 1 + - 1 b' 2 1 2 - 1 c' 1 = b' = 0 c2 [ AIEEE 2003 ] 4 = 0 are [ AIEEE 2003 ] [ AIEEE 2003 ] [ AIEEE 2003 ] ( 3. .4z = 19 is cut by m ( 16 ) The shortest distance f om the plane 12x + 4y + 3z = 327 to the sphere 2 2 2 x + y + z + 4x . 2 14 2 14 . -2 5 13 2 ( d ) 0. . 3k and b = 4 i + 3 j .y + 3z = 6 and x + y + 2z = 7 is ( a ) 0° ( b ) 30° ( c ) 45° ( d ) 60° [ AIEEE 2002 ] ( 20 ) If the lines y . Q and 1 1 1 = k. 2 j and 3 k is .k 4 i + 3 j .6k 7 → [ AIEEE 2002 ] → → → ( 22 ) A unit vector normal t (a) 6 i + 3 j + 2k → → → → → xa the plane through the points → i.12 .6 j .e → (b) i + 2 j + 3k → → → → → (c) 6 i + 3 j + 2k 7 (d) 6 i + 3 j + 2k 7 [ AIEEE 2002 ] w w w (a) 1 ( 23 ) A plane at a unit distance from the origin intersects the coordinate axes at P.5 x . then (d) 2 [ IIT 2004 ] .2 j + 6k 7 m → ra → (d) 2 i .1 z -3 = = -3 2k 2 angles. then the value of k is (a) 10 7 unit → and y .THREE DIMENSIONAL GEOMETRY ( Answers at the end of all questions ) Page 4 ( 19 ) The angle between the planes 2x .6 = = 3k 1 -5 (b) - ( c ) .10 ce .2 x .3k 7 → → (c) 3 i .6 j .3 j . R If the locus of the centroid of ∆ PQR satisfies the equation + + 2 2 x y z2 then the value of k is (b) 3 (c) 6 (d) 9 [ IIT 2005 ] ( 24 ) Two lines k is (a) 3 2 y +1 x -1 z -1 = = 2 3 4 (b) 9 2 (c) 2 9 and y -k x -3 z = = 1 2 1 intersect at a point.k 26 → → → → → is → → (a) (b) 2 i .c (d) -7 → 7 10 om → → are at right [ AIEEE 2002 ] → ( 21 ) → A vector → perpendicular → to the plane of a = 2 i .1 z . 6z = 11. 2.8z + 11 = 0 and the common circle of the spheres as the circular base of the double cone is ( a ) 24 π ( b ) 32 π ( c ) 28 π ( d ) 36 π . 6.12 .4y + z = 7. having sem . 6. 3 ) ( b ) ( 3.8y .ertical angle equal to 30° and the circular base on the plane x + y + z = 6 s c) 3 (d) 4 ( 29 ) The direct on of normal to the plane passing through origin and the line of intersection of the planes x + 2y + 3z = 4 and 4x + 3y + 2z = 1 is ( a ) ( 1.THREE DIMENSIONAL GEOMETRY ( Answers at the end of all questions ) y -2 x -1 z -k = = 1 1 2 value of k is (b) -7 (c) 1 Page 5 ( 25 ) If the line lies exactly on the plane 2x .2x 4y .c 2 There are infinite planes passing through the points ( 3. 2. b. 7 ) touching the sphere 2 2 2 x + y + z .e m ra (d) 6 ( d ) 64 ( 27 ) The mid-points of the chords cut off by th lines through the point ( 3. 7 ) 2 2 2 intersecting the sphere x + y + z . If the plane passing through th circle of contact cuts intercepts a.4y . c on the co-ordinate axes. PA × PB = ( a ) 36 ( b ) 24 ( c ) 100 om 2 2 (a) 7 ( d ) no real value [ IIT 2003 ] = 36 in points .4x .2x . 6. 1 ) ( d ) ( 3. then the ( 26 ) ( a ) 12 ( b ) 23 ( c ) 67 ( d ) 47 (a) 3 (b) 4 (c) 5 (a) 1 (b) 2 xa ( 28 ) The ratio of magnitudes of tota surface area to volume of a right circular cone with vertex at origin. 3. 2 ) w w w ( 30 ) T e volume of the double cone having vertices at the centres of the spheres 2 2 2 2 2 2 x + y + z = 25 and x + y + z . 1 ) ( c ) ( 2.6z = 11 lie on a sphere whose radius = ce . 1. then a + b + c = ( 31 ) A line through the point P ( 0. 8 ) intersects the sphere x + y + z A and B. 1 ) and ( 2. .4. 2.12 = 0 is (a) 9π ( b ) 18 π ( c ) 27 π 2 = 36 and w w w ( 37 ) A line joining the points ( 1. 4 ) ( c ) ( 3. 4.4y .6 ) 2 + y 2 + z 2 = 144 from th ( 34 ) The equation of the plane containing the line x + y through the point ( 1. 1 2 ) ( c ) ( 1. 6. . 1 ) ( d ) ( 2.6.c ( d ) 36 π 12 c 32 c 13 d 33 d 14 d 34 d 15 c 35 b ( c ) ( 4. .THREE DIMENSIONAL GEOMETRY ( Answers at the end of all questions ) 2 2 2 Page 6 ( 32 ) A sphere x + y + z .8z . 2. 4 ) ( d ) ( 3. The semi-vertical angle of the cone is ( a ) 15° ( b ) 30° ( c ) 45° ( d ) 60° is inscribed in a cone with vertex at ( 33 ) The point which is farthest on the sphere x is ( a ) ( 3. 1 ) . . 1.6z .4x . 1.y + 4 and passing om (d) 2 2 2 2 point ( 2. 6. 1 ) 2 xa (b ( 1.8 ) ( 35 ) A plane passes through the points of intersection of the spheres x + y + z = 36 2 2 2 and x + y + z . 1.12 = 0. 8. 8 ) z = 0 = 2x . 4. 6 ) ( b ) ( .8. 2 ) ( b ) ( 2. 2 ) intersects the plane x + y + z = 9 at the point ( a ) ( 3.e ( 36 ) The area of the circle formed by the intersection of the spheres x + y + z 2 2 2 x + y + z .5z = 2 (c) x + y + z = 3 ( b ) 4x + 5y .12 . 1.3.4y . 3. 4 ) .4y .5z = 4 m ra Answers 9 a 29 b 10 d 30 b 11 a 31 d ce . 6 ). 3 ) 1 a 21 c 2 c 22 c 3 b 23 d 4 b 24 b 5 b 25 a 6 c 26 d 7 c 27 a 8 b 28 c 16 a 36 c 17 a 37 d 18 a 38 19 d 39 20 a 40 . 3. A line joining the centres of the spheres intersects this plane at ( a ) ( 1.4x . 1 ) is ( a ) 3x + 4y . 2.8z .11 = 0 ( 6.2x .6z 3 ( d ) 3x + 6y .