The distance of the point ( 1 , 0 , − 3 ) from the plane x − y − z = 9 measured parallel to the line x − 2 2 = y + 2 3 = z − 6 − 6
If the plane x + 3z = 2y is rotated through a right angle about its line of intersection with the plane 2x + 3y = 4z + 5. Then the equation of the plane in its new position is
The line passing through the points ( 5 , 1 , a ) and ( 3 , b , 1 ) crosses the y z -plane at the point 0 , 17 2 , − 13 2 then
The locus of the point ( r sec α cos β , r sec α sin β , r tan α ) is
If A 1 , 2 , 3 , B 2 , 3 , 4 , C 3 , 4 , 5 , D 4 , 5 , 6 are four points, then the projection of B D ¯ on A C ¯ is
The lengths of the sides of a rectangular parallelepiped are 11,12,13. Then the angle between two diagonals out of four diagonals is
The angle between two straight lines having direction cosines l 21 , m 21 , 1 21 and 3 54 , 3 54 , 3 l 54 is 90 0 , then a pair of possible values of l , m respectively are
The equation of a line which is perpendicular to the plane 5 x + 2 y − 8 z = 0 and passing through – 1 , 0 , 1 is
The triangle whose vertices are A 3 , − 1 , 2 , B λ , − 1 , 1 , C 1 , 1 , − 2 , is a right-angled triangle. Then the sum of all possible integral values of λ is
The equation of line of intersection of two planes 4 x − 4 y − z + 11 = x + 2 y − z − 1 = 0 in symmetric form is
The equation of a line passing through the point on Z-axis at 2 units distance from origin having direction cosines ⟨ 0 , 1 , − 1 ⟩ is
The equation of the line passing through the point M ( 1 , 1 , 1 ) intersect at right angle to the line of intersection of the planes x + 2 y − 4 z = 0 and x − 1 a = y − 1 b = z − 1 c is then ⟨ a , b , c ⟩ are proportional to
The distance of the point ( 1 , 0 , − 3 ) from the plane x − y − z = 9 measured parallel to the line: x − 2 2 = y + 2 3 = z − 6 − 6
The value of m for which straight line 3 x − 2 y + z + 3 = 0 = 4 x − 3 y + 4 z + 1 is parallel to the plane 2 x − y + m z − 2 = 0 is
If the lines x − 2 1 = y − 3 1 = z − 4 − k and x − 1 k = y − 4 2 = z − 5 1 are coplanar then k can have
The direction cosines x,y,z of a line are connected by a relation ( x + y ) 2 − a 2 = 2 x y − z 2 Then a 2 + 5 a + 2019 =
If α , β , γ are the angles which a line makes with the positive direction of coordinate axes, then the value of cos 2 α − sin 2 β + cos 2 γ =
Any numbers which are proportional to the direction cosines of a line, are called
The projection of a segment joining points P ( x 1 , y 1 , z 1 ) and Q ( x 2 , y 2 , z 2 ) on a line with direction cosines l , m , n is a l + b m + c n . Then a 2 + b 2 + c 2 =
The distance of point 2 , 3 , 4 from the x-axis is k . The direction ratios of the line joining two points k + 1 , k − 3 , k , 1 , − 3 , 0 are
The direction cosines of the line passing through P − 2 , 4 , 3 and the origin are
The direction cosines of the line passing through P 1 , 2 , 5 and Q 2 , 4 , 7 are
A line makes an angle of π 4 with the positive directions of each of y – and z – axes. Then the angle that the line makes with the positive direction of x – axis, is
A line makes an angle of π 3 with the positive direction of x – axis and an angle of π 4 with the positive direction of z – axis. Then the angle that the line makes with the positive direction of y – axis, is
If O P = 10 2 and direction cosines of O P are 3 2 10 , 4 2 10 , 5 2 10 , then
If P − 2 , − 4 , − 9 and direction cosines of O P are α , β , γ , then α + β + γ × 101 is
The symmetric form of the line of intersection of two planes x + y + z + 1 = 0 , 4 x + y − 2 z + 2 = 0 is
The symmetric form of the line 2 ( x + 1 ) = y = z + 4
Direction ratios of two lines are 4 , k , 5 and 2 , − 1 , 2 . If these lines include an angle π 4 , then k can be
The angle between any two diagonals of a cube is Cos − 1 1 t . Then value of t t is
The square of the projection of the join of the points 1 , 4 , 7 and 2 , 5 , 8 on the line whose direction cosines are 1 3 , 1 3 , 1 3 , is
If the direction cosines between two lines are given by the relations a l + b m + c n = 0 and h l m + f m n + g n l = 0 , then the two lines are parallel. if a f ± b g ± c h + − 4 =
The symmetric form of the line of intersection of two planes 3 x − y = 6 , y + 2 z − 1 = 0
If a variable line in two adjacent positions has direction cosines l , m , n a n d l + δ l , m + δ m , n + δ n and δ θ is the angle between the two positions, then δ l 2 + δ m 2 + δ n 2 =
The direction ratios of two lines are given by a + b + c = 0 , a 2 + b 2 − c 2 = 0 . Then a a + c =
If A 0 , 4 , 1 , B 2 , 3 , − 1 , C 4 , 5 , 0 respectively, then the angle between A B , B C is α 2 where α =
If P and Q are complementary angles and P , Q , R are angles made by a straight line with X – , Y – and Z – axes respectively, then R =
If O is the origin and the line O P of length r makes an angle α with x – axis and lies in the x z – plane, then the coordinates of P are
If the vertices of a triangle are A 1 , 2 , 1 , B 4 , 3 , 1 , C 3 , 1 , 5 then Area of Δ A B C 42 × cos 2 A =
The foot of the perpendicular from P 5 , 7 , 3 to the line which is passing through the points A 12 , 21 , 10 , B 9 , 13 , 15 is
If the direction cosines l , m , n of two perpendicular lines are connected by the relations λ l + 3 m − 5 n = 0 and 10 l 2 + m 2 − 5 n 2 = 0 then λ 5 is
Triangle formed by three lines whose direction ratios are − 1 , 1 , 1 , 1 , 2 , 0 , 1 , 1 , 0 is
Let a line makes an angle θ with y – , z – axes and ϕ with x – axis. If sin ϕ and sin θ are in the ratio m : n , then cos θ =
The condition that the straight lines x = a 1 y + b 1 ; z = c 1 y + d 1 and x = a 2 y + b 2 ; z = c 2 y + d 2 are perpendicular, is
The direction ratios of two lines are given by p , q , r and 1 q r , 1 r p , 1 p q . Ifis the angle between them such that 0 ≤ α < π , then the value of cos α + cos 2 α + cos 3 α + …. + cos p α ⋅ cos q α ⋅ cos r α =
If the direction cosines of two lines are connected by the relations l + 3 m + 5 n = 0 and 2 l 2 − 5 m 2 + n 2 = 0 , then the sum of all possible values of m n is
The perpendicular bisector of the line segment with end points 2 , 3 , 2 and − 4 , 1 , 4 passes through the point − 3 , 6 , 1 and has equation of the form x + 3 a = y − 6 b = z − 1 c where a , b , c are relatively prime integers with a > 0 , then the value of a b c − a + b + c is
The equation of the line passing through the points ( 2 , 3 , 5 ) , ( 1 , − 2 , 1 )
Let ‘ L ’ be the line of intersection of the planes 2 x + 3 y + z = 1 and x + 3 y + 2 z = 2 , if L makes an angle α with positive x – axis, then cos α equals.
A line with direction ratios 2 , 1 , 2 intersect each of the lines x = y + a = z and x + a = 2 y = 2 z the co-ordinates of each of the points of intersection are
If the equation of plane containing lines x − 1 2 = y + 1 − 1 = z − 1 3 and x − 3 − 4 = z − 2 2 = z − 1 − 6 is 9 x + b y + c z = d then b + c + d is
The equation of the line passing through the point 1 , 2 , 6 and parallel to y – axis is
The equation of a line passing through the point 1 , 1 , 1 and parallel to y z – plane is
The equation of a line passing through the point on x – axis at a distance of 3 units from the origin and parallel to x y – plane is
Equation of line parallel to the line x − 4 2 = y + 1 − 3 = z + 10 8 and passing through the point − 1 , 2 , 3 is
Equation of line parallel to the line x = y = z and passing through the point 1 , 0 , 0 is
The equation of a line passing through the point on z – axis at a distance of 1 unit from the origin and having direction ratios 1 , 0 , − 1 is
The equation of the line passing through the points 1 , 2 , 3 , 2 , − 1 , 2 is
The number of lines perpendicular to y – axis and lying in x z – plane is
The value of t for which the line x − 1 2 = y − 3 t = z − 1 1 passes through the point 1 , 3 , 1 and 3 , 1 , 2 is
Equation of line passing through 1 , − 1 , 0 and parallel to the line having direction cosines 3 2 , 0 , 1 2 is
The line x − 2 3 = y − 3 4 = z − 4 5 is parallel to the plane
Which of the following lines lies on the plane x + 2 y – z + 4 = 0
The equation of the line passing through ( 1 , – 1 , 2 ) and parallel to the line of intersection of the planes 2 x − y + z + 1 = 0 and x − y + z + 2 = 0
If the foot of the perpendicular drawn from the point (1,0,3) on a line passing though α , 7 , 1 is 5 3 , 7 3 , 17 3 , then α is equal to
A line passes through the point A ( i ^ + 2 j ^ + 3 k ^ ) and is parallel to the vector V ( i ^ + j ^ + k ^ ) . The shortest distance from the origin, to the line is……
The value of m for which straight line 3 x – 2 y + z + 3 = 0 = 4 x – 3 y + 4 z + 1 is parallel to the plane 2 x – y + m z – 2 = 0 is
The plane passing through the point (-2, -2, 2) and containing the line joining the points (1, 1, 1) and (1, -1, 2) makes intercepts on the coordinate axes, the sum of whose lengths is
The plane P 1 : 4 x + 7 y + 4 z + 81 = 0 is rotated through a right angle about its line of intersection with the plane P 2 : 5 x + 3 y + 10 z − 25 = 0 . If P 3 = 0 is the equation of plane P 1 in its new position and if k is the distance of P 3 = 0 from origin, then k =
Equation of a line in the plane π ≡ 2 x − y + z − 4 = 0 which is perpendicular to the line l whose equation is x − 2 1 = y − 2 − 1 = z − 3 − 2 and which passes through the point of intersection of l and π is
A plane which is perpendicular to two planes 2 x − 2 y + z = 0 and x − y + 2 z = 4 passes through ( 1 , − 2 , 1 ) . The distance of the plane from the point (1, 2 , 2 ) is
The equation of line which is parallel to the line x − 2 3 = y + 1 1 = z − 1 9 and also passing through the point ( 3 , 0 , 5 ) is
Equation of line passing through the point ( 1 , 0 , 3 ) and parallel to a line whose direction ratios are ⟨ 3 , 6 , 2 ⟩ is
Equation of line passing through the point ( 0 , 0 , 0 ) and equally inclined to the coordinate axes is
For the line x − 1 1 = y − 2 2 = z − 3 3 which one of the following is incorrect.
The equation of the line passing through the point ( a , b , c ) and parallel to z − axis is
The equation of the line passing through the point ( 0 , 0 , 0 ) and parallel to x − axis is
Equation of a line passing through – 1 , 2 , – 3 and perpendicular to the plane 2 x + 3 y + z + 5 = 0 is
The direction cosines of the line of intersection of two planes x − y + z − 5 = 0 , x − 3 y − 6 = 0 are
The direction cosines of the line x − 1 l = y + 1 l + 1 = z − 1 l
If the points A ( − 1 , 0 , 7 ) , B ( 3 , 2 , k ) , C ( 5 , 3 , − 2 ) are collinear then k =
The equation of line parallel to the plane x + y + z = 2 and intersecting the line x + y − z = 0 = x + 2 y − 3 z + 5 at the point – 4 , 1 , 3 is
Equation of a line passing through – 1 , 2 , – 3 and perpendicular to the plane 2 x + 3 y + z + 5 = 0 is
The symmetric form of the line of intersection of two planes x − y + 2 z = 5 , 3 x + y + z = 7 is
The symmetric form of the line x = a y + b , and z = c y + d
The symmetric form of the line x + y – z = 1 and 2 x – 3 y + z = 2 is
The direction angles made by a line ‘ L ’ with the coordinate axes, are α , β , γ . Then the range of sin 2 α + sin 2 β
Suppose that l , m , n are direction cosines of a line ” L ” If the direction angles of the line are replaced by their supplements, the new direction cosines are
If a line passing through origin is inclined at 60 ° and 45 ° to X – and Y – axes respectively, Then the angle at which it is inclined to Z – axis, is
If l 1 , m 1 , n 1 and l 2 , m 2 , n 2 are the direction cosines of the two lines inclined to each other at an angle θ , then the direction cosines of the external angular bisector of the angle between these lines are l 1 − l 2 2 sin α , m 1 − m 2 2 sin α , n 1 − n 2 2 sin α where 2 α =
A line makes angles α , β , γ , δ with the diagonals of a cube. Then ∑ cos 2 α = 1 λ where λ =
If O P = 10 2 and direction cosines of O P are 3 2 10 , 4 2 10 , 5 2 10 , then
If 2 , 3 , 4 and 1 , 2 , 3 are direction ratios of O A and O B , then the direction cosines of a line perpendicular to both the lines O A and O B are
The direction ratios of the line whose direction angles are π 3 , π 3 , π 4 , are
The number of lines which are equally inclined to the coordinate axes, is k. If k 2 + e log e 6 A + 1 3 ( k ) = 4 k . Then k − A =
If 1 , − 1 , 1 and 2 , − 3 , 5 are direction ratios of two lines, then the angle between them is
The symmetric form of the line of intersection of planes 3 x + 2 y + z − 5 = 0 and x + y − 2 z = 0
The symmetric form of the line passing through a point 1 , 0 , 3 and having direction ratios ⟨ 1 , 0 , 3 ⟩ is
The vertices of a triangle are A ( 1 , 0 , 2 ) , B ( 2 , 3 , 0 ) , C ( 0 , 0 , 4 ) then the symmetric form of the line passing through centroid of the triangle and one vertex B is
A line makes angles α , β , γ with coordinate axes. If β + γ = π 2 , then cos α + cos β + cos γ 2 =
A line makes angles α , β , γ with positive axes, then the range of ∑ sin α sin β is
A 1 , 2 , 3 , B 4 , 5 , 7 , C − 4 , 3 , − 6 , D 2 , k 2 , 2 are four points. If the lines A B and C D are parallel, then k =
If the direction ratios of two perpendicular straight lines are given by 1 , θ , θ 3 − θ + 7 and 0 , 4 , θ where θ ≠ 0 , then Tan − 1 θ − θ 3 11 =
The direction ratios of the line x = p y + q , z = r y + s are
A = − 1 , 2 , − 3 , B = 5 , 0 , − 6 , C = 0 , 4 , − 1 . If l , m , n be the direction cosines of the internal angular bisector of ∠ B A C , then l − m − n =
If O P is equally inclined to O X , O Y and O Z and if P is 3 units from the origin then P which lies in the first octant is
The foot of the perpendicular from 2 , − 4 , 0 to the join of 0 , 0 , 0 , 0 , 3 , 0 is
The angle between two straight lines is α . One line has direction cosines 1 2 , 1 2 , 1 2 and the other line has direction ratios 0 , 1 , 2 . Then tan α =
If the direction cosines of two lines are connected by the relations a l + b m + c n = 0 and u l 2 + v m 2 + w n 2 = 0 , then the two lines are parallel if
The acute angle between the two lines whose direction ratios are connected by l + m − n = 0 a n d l 2 + m 2 − n 2 = 0 is
If the direction cosines of two lines are connected by the relations a l + b m + c n = 0 and u l 2 + v m 2 + w n 2 = 0 , then the two lines are parallel if
The direction ratios of two lines are given by a + b + c = 0 and a b + b c − 2 c a = 0 . Then the angle between the lines is
The distance of a point A − 2 , 3 , 1 from the line M N through M − 3 , 5 , 2 , which makes equal angles with the coordinate axes is
If r ¯ is a vector of magnitude 210 and has direction ratios 2 , − 3 , 6 , then r ¯ =
If a line makes angles α , β , γ with coordinate axes respectively, then the value of ∑ cos 2 α ∑ cos 2 α + ∑ sin 2 α =
The direction cosines of the lines bisecting the angles between two concurrent lines having direction ratios 1 , 2 , 2 and 6 , − 2 , − 3 are
P 0 , 4 , − 1 , Q − 1 , 2 , − 3 and R 5 , 0 , − 6 are the vertices of a triangle. The direction ratios of the internal angular bisector of ∠ P Q R are
A line makes angles 60 ° , 60 ° , 45 ° and θ with the four diagonals of a cube. Then sin 2 θ =
If O 0 , 0 , 0 , A 2 , 3 , − 4 , B 3 , 4 , − 5 and cos ∠ O A B = p + 2 Q − 3 , then p Q + 4 =
If P 4 , − 1 , 3 , Q 0 , 6 , 7 , R − 2 , 1 , 9 are vertices of a triangle P Q R ,the equation of the median through R is x − i m = y − c s = z − t y , then m + y + s + t + i + c = .(where[.] denotes greatest integer function) and m , s , y are numerically least possible integers.
The direction cosines of three mutually perpendicular straight lines are l 1 , m 1 , n 1 , l 2 , m 2 , n 2 , l 3 , m 3 , n 3 . Then the direction cosines of a line which is equally inclined to the given three lines, are
If 3 , − 1 , 2 , 4 , 1 , 0 are the direction ratios of two straight lines, then which of the following are not direction numbers perpendicular to both these lines?
If a line makes acute angles α , β , γ with the coordinate axes such that cos α cos β = 1 6 ; cos β cos γ = 1 2 3 ; cos γ cos α = 1 3 2 then , 3 cos α + cos β + cos γ 2 = 2 + 3 + 6 + λ then λ =
If the direction cosines of two lines are given by l + m + n = 0 a n d 1 l − 2 m + 1 n = 0 , then the angle between them is
The length of the projection of the line segment joining the points − 1 , − 2 , 2 and 1 , 0 , 2 on the plane x + 3 y − 5 z − 2 = 0 is equal to
The equation of line passing through ( 1 , 5 , 3 ) and having direction ratios ⟨ l , m , 1 2 ⟩ in symmetric form is x − 1 l = y − 5 m = z − 3 1 2 then l 2 + m 2 =
A line with positive direction cosines passes through the point A 1 , 2 , − 1 and makes equal angles with the coordinate axes. The line meets the plane x + 2 y + z = 5 at the point B . Then the point B and length of the line segment A B ¯ respectively, are equal to
Equation of a line through ( 5,3,6 ) and parallel to z − axis is
The equation of straight line parallel to 2 i ¯ − j ¯ + 3 k ¯ and passing through the point (5,-2,4) is
The equation of straight line passing through the point ( a , b , c ) and parallel to z – axis is
Equation of the line x − 1 2 = y + 1 − 1 = z − 3 4 is the line of intersection of ——-
The parametric form of the equation of the line 3 x – 6 y – 2 z – 15 = 0 = 2 x + y – 2 z – 5 is
Which of the following is correct to the line 5 x + y + 3 z = 0 and 3 x − y + z + 1 = 0
Equation of the plane through the lines x − 1 2 = y + 1 − 2 = z − 3 1 and x − 1 1 = y + 1 2 = z − 3 2 is
If the line x − 2 3 = y − 1 − 5 = z + 2 2 lies in the plane x + 3 y − α z + β = 0 then α , β =
The point on the line x + 2 2 = y + 6 3 = z − 34 − 10 which is nearest to the line x + 6 4 = y − 7 − 3 = z − 7 − 2 is a , b , c where a + b + c =
The lines x − y + z = 1 , 3 x − y − z + 1 = 0 and x + 4 y − z = 1 , x + 2 y + z + 1 = 0 are
Let L be the line passing through the point P 1 , 2 , 0 and perpendicular to the plane r ¯ ⋅ 3 i ¯ + 4 k ¯ = 0 if line intersects the plane r ¯ ⋅ i ¯ − j ¯ + k ¯ = 13 at point Q , then Q is
Which of the following is correct to the line x 2 = y 3 = z 5
Equation of line passing through the point 1 , 0 , − 3 and parallel to a line whose direction ratios are 1 , 2 , 2 is
Equation of line passing through the point 0 , 0 , 0 and parallel to a line whose direction ratios are 2 , 3 , 4 is
The equation of line passing through the points 3 , p , 0 and 2 , q , 4 where p + q = 5 and p q = 6 is p > q
The number of lines perpendicular to x – axis and lying in y z – plane is
The equation of the line joining the points − 2 , 4 , 2 and 7 , − 2 , 5 is
Equation of line passing through the point a , b , c and making equal angles with coordinate axes where a , b , c are all positive
Equation of the line passing through the point 1 2 , 0 , 1 and making angles 90 ° , 45 ° , 45 ° with axes respectively is
Equation of the line passing through the point 1 2 , 0 , 1 and making angles 90 ° , 45 ° , 45 ° with axes respectively is
Equation of the line passing through the point − 1 , 2 , − 3 and perpendicular to the plane 2 x + 3 y + z + 5 = 0
Let L be the line of intersection of the planes 2 x + 3 y + z = 1 and x + 3 y + 2 z = 2 . If L makes an angle α with the positive x – axis then cos α is
The distance between the points 5 , 1 , 7 and c , − 5 , 1 is 9 then c =
The equation of the passing through ( 1 , 0 , 3 ) and having direction cosines a , b , 1 2 in Symmetric form is x − 1 a = y b = z − 3 1 2 then a 2 + b 2 is
The symmetric form of the line whose parametric equations are given by x = 1 – 2 t , y = t , z = – 1 + t is
The line of intersection of the planes x + y + z – 1 = 0 , 4 x + y – 2 z + 2 = 0 passes through the point
The line of intersection of the planes x – 2 y + z – 3 = 0 a n d x + y – 2 z + 3 = 0 is parallel to the line
The two lines x = a y + b , z = c y + d and x = a ‘ y + b ‘ , Z = c ‘ y + d ‘ will be perpendicular if and only if
The parametric equation of the line of intersection of the given planes are 3 x – 6 y – 2 z = 15 , and 2 x + y – 2 z = 5 are
The parametric equation of the line of intersection of the given planes are 3 x – 6 y – 2 z = 15 , and 2 x + y – 2 z = 5 are
The line of intersection of the planes x – y + 6 z = 0 and 2 x + 3 y – z = 5 is
The equation of the line passing through ( 1 , – 1 , 3 ) which is perpendicular to the plane 9 x – 2 y – 5 z + 4 = 0 is
The equation of line which is perpendicular to the plane 2 x – y + 6 z = 0 and passing through ( 1 , 0 , 1 ) is
The equation of the line having equal inclinations with coordinate axes and passing through one point in line of intersection of the planes 2 x − y + z = 2 and x − y − z = − 1
A line having direction ratios ⟨ 3 , 4 , 5 ⟩ cuts two planes 2 x − 3 y + 6 z − 12 = 0 and 2 x − 3 y + 6 z + 2 = 0 at point P and at point Q then the length of PQ
A line having direction ratios ⟨ 1 , 0 , 3 ⟩ cuts planes x – y + 6 z = 6 and x – y + 6 z = 4 at P and Q then P Q .
Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0,1) and R be any point (2, 1,6). Then the image of R in the plane P is:
The shortest distance between the lines x – 3 3 = y – 8 – 1 = z – 3 1 and x + 3 – 3 = y + 7 2 = z – 6 4 is:
The distance of the point ( 1 , 0 , – 3 ) from the plane x – y – z = 9 measured parallel to the line x − 2 2 = y + 2 3 = z − 6 − 6 i s
Given lines x – 4 2 = y + 5 4 = z – 1 – 3 and x – 2 1 = y + 1 3 = z 2 Statement S 1 : The lines intersect Statement S 2 : They are not parallel
The distance of the point – i ¯ + 2 j ¯ + 6 k ¯ from the straight line through the point ( 2 , 3 , – 4 ) and parallel to 6 i ¯ + 3 j ¯ – 4 k ¯ is
A plane passing through 1 , 1 , 1 intersects the positive direction of coordinate axes at A , B and C then the volume of the tetrahedron O A B C satisfies
If the reflection of the point P ( 1 , 0 , 0 ) in the line x − 1 2 = y + 1 − 3 = z + 10 8 is ( α , β , γ ) , then α + β + γ i s
T h e d i s tan c e o f t h e z – a x i s f r o m t h e i m a g e o f t h e p o i n t M ( 2 , – 3 , 3 ) i n t h e p l a n e x – 2 y – z + 1 = 0 , i s
If the points A ( 2 − x , 2 , 2 ) , B ( 2 , 2 − y , 2 ) , C ( 2 , 2 , 2 − z ) and D ( 1 , 1 , 1 ) are coplanar, then locus of P ( x , y , z ) is
The distance of the point ( 1 , − 2 , 3 ) from the plane x − y + z − 5 = 0 , parallel to the line x 2 = y 3 = z − 1 − 6 is
The distance of the plane passing through the point P ( 1 , 1 , 1 ) and perpendicular to the line x − 1 3 = y − 1 0 = z − 1 4 from the origin is
Angle between the two planes of which one plane is 4 x + y + 3 z = 0 and another plane containing the lines x − 3 2 = y − 2 3 = z − 1 3 , x − 2 3 = y − 3 2 = z − 2 3 is
If the reflection of the point P ( 1 , 0 , 0 ) in the line x − 1 2 = y + 1 − 3 = z + 10 8 is ( a , b , c ) then | a + b + c | 7 =
If the distance between the planes x – 2y + z = d and the plane containing the lines x − 1 2 = y − 2 3 = z − 3 4 and x − 2 3 = y − 3 4 = z − 4 5 is 6 , then | d | is
The image of the point ( 1 , 2 , − 1 ) , on the plane containing the line x + 1 − 3 = y − 3 2 = z + 2 1 and the point ( 0 , 7 , − 7 ) , is
The vector equation of the plane passing through the line of intersection of two planes x − 2 y + 3 z − 1 = 0 , 2 x + y + z − 2 = 0 and passing through the point ( 1 , 2 , 3 ) is
If the line x − 2 3 = y − 1 − 5 = z + 2 2 lie in the plane x + 3 y − α z + β = 0 then 1 2 α 2 + β 2 =
Equation of the plane such that foot of altitude drawn from ( − 1 , 1 , 1 ) to the plane has the coordinate ( 3 , − 2 , − 1 ) is
Let A (1, 1, 1), B (2,3,5) and C (-1 ,0,2) be three points, then equation of a plane parallel to the plane ABC which is at distance 2 is
Let the equations of a line and a plane be x + 3 2 = y − 4 3 = z + 5 2 and 4 x − 2 y − z = 1 , respectively, then
The value of k such that x − 4 1 = y − 2 1 = z − k 2 lies in the plane 2 x − 4 y + z = 7 is
Let P ( 3 , 2 , 6 ) be a point in space and Q be a point on line r = ( i ^ − j ^ + 2 k ^ ) + μ ( − 3 i ^ + j ^ + 5 k ^ ) . Then the value of μ for which the vector P Q is parallel to the plane x − 4 y + 3 z = 1 is
The ratio in which the plane r ⋅ ( i − 2 j + 3 k ) = 17 divides the line joining the points − 2 i + 4 j + 7 k and 3 i − 5 j + 8 k is
The image of the point ( -1,3,4) in the plane x-2y -0 is
The point of intersection of the line passing through ( 0 , 0 , 1 ) and intersecting the lines x + 2 y + z = 1 , − x + y − 2 z = 2 and x + y = 2 , x + z = 2 with x y plane is
The plane which passes through the point ( 3 , 2 , 0 ) and the line x − 3 1 = y − 6 5 = z − 4 4 is
Given α = 3 i ^ + j ^ + 2 k ^ and β = i ^ − 2 j ^ − 4 k ^ are the position vectors of the points A and B . Then the distance of the point − i ^ + j ^ + k ^ from the plane passing through B and perpendicular to AB is
The three planes 4 y + 6 z = 5 , 2 x + 3 y + 5 z = 5 and 6 x + 5 y + 9 z = 10
The equation of the plane through the line of intersection of the planes ax + by + cz + d = 0 and a ′ x + b ′ y + c ′ z + d ′ = 0 and parallel to the line y = 0 and z = 0 is
The vector equation of the plane passing through the origin and the line of intersection of the planes r . a = λ and r ⋅ b = μ is
From the point P(a, b, c),let perpendiculars PL and PM be drawn to YOZ and ZOX planes, respectively. Then the equation of the plane OLM is
The intercepts made on the axes by the plane which bisects the line joining the points (1, 2, 3) and (-3, 4, 5) at right angles are
The plane 4 x + 7 y + 4 z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5 x + 3 y + 10 z = 25 . The equation of the plane in its new position is
The nature of the intersection of the set of planes x + ay + (b + c) z + d= 0, x + by +(c + a)z + d -0 and x+ cy + (a+b)z +d=0?
The locus represented by x y + y z = 0 is a pair of
If the line x − 2 − 1 = y + 2 1 = z + k 4 is one of the angle bisectors of the lines x 1 = y − 2 = z 3 and x − 2 = y 3 = z 1 then the value of k is
What is the equation of the plane which passes through the e-axis and is perpendicular to the line x − a cos θ = y + 2 sin θ = z − 3 0 ?
The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, -1, 4) with the plane 5x – 4y – z- 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is
The projection of the line x + 1 − 1 = y 2 = z − 1 3 on the plane x − 2 y + z = 6 is the line of intersection of this plane with the plane
The variable plane ( 2 λ + 1 ) x + ( 3 − λ ) y + z = 4 always passes through the line
If the projection of the line x 2 = y − 1 2 = z − 1 1 on a plane P is x 1 = y − 1 1 = z − 1 − 1 . Then the distance of plane P from origin is
The line x − 2 3 = y − 1 2 = z − 1 − 1 intersects the curve x 2 − y 2 = a 2 ; z = 0 then absolute value of a is equal to
If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of Δ A B C is
Distance between two parallel planes 2 x + y + 2 z = 8 and 4 x + 2 y + 4 z + 5 = 0 is
An equation of a plane parallel to the plane x – 2 y + 2 z – – 5 = 0 and at a unit distance from the origin is
If the lines x − 1 2 = y + 1 3 = z − 1 4 and x − 3 1 = y − k 2 = z 1 intersect then k =
If the line x − 2 3 = y + 1 2 = z − 1 − 1 intersects the plane 2 x + 3 y – z + 13 = 0 at a point P then PQ is equal to
The distance of the point (1,0,2) from the point of intersection of the line x − 2 3 = y + 1 4 = z − 2 12 and the plane x − y + z = 16 is
If the angle between the line x = y − 1 2 = z − 3 λ and the plane x + 2 y + 3 z = 4 is cos − 1 ( 5 / 14 ) then λ =
The distance of the point (1,-5,9) from the plane x – y + z = 5 measured along a straight line x = y = z is
If the image of the point P ( 1 , − 2 , 3 ) in the plane, 2 x + 3 y − 4 z + 22 = 0 measured parallel to the line, x 1 = y 4 = z 5 is Q, then PQ is equal to
The distance of the plane 3 x + 4 y + 5 z + 19 = 0 from the point (1,-1,1) measured along a line parallel to the line with direction ratios 2,3,1 is
The image of the line x − 1 3 = y − 3 1 = z − 4 − 5 in the plane 2 x − y + z + 3 = 0 is the line
The coordinates of a point on the line x = 4 y + 5 , z = 3 y – 6 at a distance 3 26 from the point ( 5 , 0 , − 6 ) are
The algebraic sum of the intercepts made by the plane x − 2 y + 3 z = 24 on the coordinate axes is
If the points ( 1 , 1 , p ) and ( – 3 , 0 , 1 ) are equidistant from the plane r ( 3 i + 4 j – 12 k ) + 13 = 0 then the value of 3 p is ( 3 p > 4 )
l = m = n = 1 represents the direction ratios of
An equation of the plane passing through the line of intersection of the planes x + y + z = 6 and 2 x + 3 y + 4 z + 5 = 0 and passing through ( 1 , 1 , 1 ) is
The volume of the tetrahedron included between the plane 3 x + 4 y – 5 z – 60 = 0 and the coordinate planes in cubic units is
If θ denotes the acute angle between the line r = ( i + 2 j – k ) + A ( i – j + k ) and the plane r . ( 2 i – j + k ) = 4 , then sin θ + 2 cos θ =
The shortest distance between the lines r = ( 4 i − j ) + λ ( i + 2 j − 3 k ) and r = ( i − j + 2 k ) + μ ( 2 i + 4 j − 5 k ) is
