12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d )

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1 - THREE DIMENSIONAL GEOMETRY Page ( ) If the angle θ between the line x - y + x + y - z - and the plane λ x is such that sin θ, then the value of λ is [ AIEEE 00 ] ( ) If the plane ax - ay + 4az passes through the midpoint of the line joining the centres of the spheres x + y + z + 6x - 8y - z and x + y + z - 0x + 4y - z 8, then a equals - - [ AIEEE 00 ] ( ) The distance between the line r plane r. ( ^ i + ^j + ^k ) s 0 0 c 0 ^i - ^j + ^k + λ ( ^i - ^j + 4 ^k ) and the ( 4 ) The angle between the lines x y - z and 6x - y - 4z is 0 [ AIEEE 00 ] 0 b ) [ AIEEE 00 ] ( ) The p ane x + y - z 4 cuts the sphere x + y + z - x + z - 0 in a circle of radius [ AIEEE 00 ] ( 6 ) A line makes the same angle θ with each of the X- and Z- axis. If the angle β, which it makes with the y-axis, is such that sin β sin θ, then cos θ equals [ AIEEE 004 ]

2 - THREE DIMENSIONAL GEOMETRY Page ( ) Distance between two parallel planes x + y + z 8 and 4x + y + 4z + 0 is ( 8 ) A line with direction cosines proportional to,, meets each o the nes x y + a z and x + a y z. The coordinates of each of the p ints of intersection are given by [ AIEEE 004 ] ( a, a, a ), ( a, a, a ) ( a, a, a ), ( a, a, a ) ( a, a, a ), ( a, a, a ) ( a, a, a ), ( a, a, a ) [ AIEEE 004 ] ( ) If the straight lines x + s, y - - λs, z + λs and x t, y + t, z - t, with parameters s and t respectively, are co-planar, then λ equals [ AIEEE 004 ] ( 0 ) The intersection of the sphe es x + y + z + x - y - z and x + y + z - x + y + 4z 8 is the same as the intersection of one of the spheres and the plane x - y - z x - y - z x - y z x - y - z [ AIEEE 004 ] ( ) The ines x ay + b, z cy + d and x a y + b, z c y + d will be perpendicular if and only if aa + cc + 0 aa + cc 0 aa + bb 0 and aa + bb + cc 0 [ AIEEE 00 ] ( ) The lines x - y - z k and x - k y - 4 z - are coplanar, if k 0 or - k or - k 0 or - k or - [ AIEEE 00 ]

3 - THREE DIMENSIONAL GEOMETRY Page ( ) Two systems of rectangular axes have the same origin. If a plane cuts them at distances a, b, c and a, b c from the origin, then a b c a' b' c' a b c a' b' c' a b c a' b' c' a b c a' b' c 0 [ AIEEE 00 ] ( 4 ) The direction cosines of the normal to the plane x + y - z 4 0 are -, -, ,, 4 4 4, 4, 4, 4 4, - [ AIEEE 00 ] 4 ( ) The radius of a circle in which the sphe e x + y + z + x - y - 4z is cut by the plane x + y + z + 0 is 4 [ AIEEE 00 ] ( 6 ) The shortest distance f om the plane x + 4y + z to the sphere x + y + z + 4x - y - 6z is 6 [ AIEEE 00 ] ( ) The dist nce of a point (, -, ) from the plane x - y + z and parallel to the x y z line is - 6 [ AIEEE 00 ] ( 8 ) The co-ordinates of the point in which the line joining the points (,, - ) and ( -,, 8 ) and intersected by the YZ-plane are 0,, 0, -, - 0, -, 0,, [ AIEEE 00 ]

4 - THREE DIMENSIONAL GEOMETRY Page 4 ( ) The angle between the planes x - y + z 6 and x + y + z is [ AIEEE 00 ] x - y - ( 0 ) If the lines - k angles, then the value of k is z - and x - k y - z - 6 are at right [ AIEEE 00 ] 0 ( ) A unit vector perpendicular to the plane of a i - 6 j - k and b 4 i + j - k is 4 i + i j 6 - j + - k 6 k i i - 6 j - - j - k 6 k ( ) A unit vector normal t the plane through the points 6 i + j + k 6 i + j + k i + j + k 6 i + j + k [ AIEEE 00 ] i, j and k is [ AIEEE 00 ] ( ) A plane at a unit distance from the origin intersects the coordinate axes at P, Q and R If the locus of the centroid of PQR satisfies the equation + + k, x y z then the value of k is 6 [ IIT 00 ] ( 4 ) Two lines k is x - y + z - 4 and x - y - k z intersect at a point, then [ IIT 004 ]

5 - THREE DIMENSIONAL GEOMETRY Page x - ( ) If the line value of k is y - z - k lies exactly on the plane x - 4y + z, then the - no real value [ IIT 00 ] ( 6 ) There are infinite planes passing through the points (, 6, ) touching the sphere x + y + z - x - 4y - 6z. If the plane passing through th circle of contact cuts intercepts a, b, c on the co-ordinate axes, then a + b + c 6 4 ( ) The mid-points of the chords cut off by th lines through the point (, 6, ) intersecting the sphere x + y + z - x 4y - 6z lie on a sphere whose radius 4 6 ( 8 ) The ratio of magnitudes of tota surface area to volume of a right circular cone with vertex at origin, having sem - ertical angle equal to 0 and the circular base on the plane x + y + z 6 s c ) 4 ( ) The direct on of normal to the plane passing through origin and the line of intersection of the planes x + y + z 4 and 4x + y + z is (,, ) (,, ) (,, ) (,, ) ( 0 ) T e volume of the double cone having vertices at the centres of the spheres x + y + z and x + y + z - 4x - 8y - 8z + 0 and the common circle of the spheres as the circular base of the double cone is 4 π π 8 π 6 π ( ) A line through the point P ( 0, 6, 8 ) intersects the sphere x + y + z 6 in points A and B. PA PB

6 - THREE DIMENSIONAL GEOMETRY Page 6 ( ) A sphere x + y + z - x - 4y - 6z - 0 is inscribed in a cone with vertex at ( 6, 6, 6 ). The semi-vertical angle of the cone is ( ) The point which is farthest on the sphere x + y + z 44 from th point (, 4, 4 ) is (, 6, 6 ) ( -, - 6, - 6 ) ( 4, 8, 8 ) - 4, - 8, - 8 ) ( 4 ) The equation of the plane containing the line x + y z 0 x - y + 4 and passing through the point (,, ) is x + 4y - z 4x + y - 6z x + y + z x + 6y - z 4 ( ) A plane passes through the points of intersection of the spheres x + y + z 6 and x + y + z - 4x - 4y - 8z - 0. A line joining the centres of the spheres intersects this plane at (,, ) ( b (, ) (,, ) (,, ) ( 6 ) The area of the circle formed by the intersection of the spheres x + y + z 6 and x + y + z - 4x - 4y - 8z - 0 is π 8 π π 6 π ( ) A line joining the points (,, ) and (,, ) intersects the plane x + y + z at the point (, 4, ) (,, 4 ) (,, 4 ) (,, ) Answers a c b b b c c b a d a c d d c a a a d a c c d b a d a c b b d c d d b c d

If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2)

If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2) Sphere Definition: A sphere is the locus of a point which remains at a constant distance from a fixed point. The fixed point is called the centre and the constant distance is the radius of the sphere.