Equation of Line in 3D: JEE Questions & Solutions

Let (α, β, γ) be the co-ordinates of the foot of the perpendicular drawn from the point (5, 4, 2) on the line r=(-i+3j+k)+λ(2i+3j-k). Then…

Let the line L₁ be parallel to the vector -3i + 2j + 4k and pass through the point (2, 6, 7), and the line L₂ be parallel to the vector 2i…

Let the line L pass through the point (-3, 5, 2) and make equal angles with the positive coordinate axes. If the distance of L from the…

If the image of the point P(1, 2, a) in the line (x - 6)/3 = (y - 7)/2 = (7 - z)/2 is Q(5, b, c), then a² + b² + c² is equal to

Let L be the line (x+1)/2 = (y+1)/3 = (z+3)/6 and let S be the set of all points (a,b,c) on L, whose distance from the line (x+1)/2 =…

The vertices B and C of a triangle ABC lie on the line x/1 = (1-y)/(-2) = (z-2)/3. The coordinates of A and B are (1, 6, 3) and (4, 9, α)…

If the image of the point P(a, 2, a) in the line x/2 = (y+a)/1 = z/1 is Q and the image of Q in the line (x-2b)/2 = (y-a)/1 = (z+2b)/(-5)…

Let the lines L₁: r = i + 2 j + 3 k + λ(2 i + 3 j + 4 k), λ ∈ ℝ and L₂: r = (4 i + j) + μ(5 i + + 2 j + k), μ ∈ ℝ intersect at the point R.…

Let a line L passing through the point P(1,1,1) be perpendicular to the lines (x-4)/4=(y-1)/1=(z-1)/1 and (x-17)/1=(y-71)/1=z/0. Let the…

If the distances of the point (1,2,a) from the line (x-1)/1=y/2=(z-1)/1 along the lines L₁: (x-1)/3=(y-2)/4=(z-a)/b and L₂:…

Let P be a point in the plane of the vectors AB=3i+j-k and AC=i-j+3k such that P is equidistant from the lines AB and AC. If |AP|=√5/2,…

Let Q(a,b,c) be the image of the point P(3,2,1) in the line (x-1)/1=(y-2)/2=(z-1)/1. The distance of Q from the line…

If the distance of the point P(43,α,β), β<0, from the line r = 4 i- k+μ(2 i+3 k), μ∈ℝ along a line with direction ratios 3,-1,0 is 13√10,…

Let L₁: (x - 1)/3 = (y - 1)/(-1) = (z + 1)/0 and L₂: (x - 2)/2 = y/0 = (z + 4)/α, where α ∈ ℝ, be two lines which intersect at the point B.…

The perpendicular distance of the line (x - 1)/2 = (y + 2)/(-1) = (z + 3)/2 from the point P(2, -10, 1) is:

Let a line pass through two distinct points P(-2, -1, 3) and Q, and be parallel to the vector 3i + 2j + 2k. If the distance of the point Q…

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line: (x-3)/7 = (y-2)/(-1) = (z+1)/(-2) Then the area of the…

The distance of the line (x - 2)/2 = (y - 6)/3 = (z - 3)/4 from the point (1, 4, 0) along the line x/1 = (y - 2)/2 = (z + 3)/3 is:

Let in a △ ABC, the length of the side AC is 6, the vertex B is (1, 2, 3) and the vertices A, C lie on the line (x - 6)/3 = (y - 7)/2 = (z…

Let the line passing through the points (-1, 2, 1) and parallel to the line (x - 1)/2 = (y + 1)/3 = z/4 intersect the line (x + 2)/3 = (y -…

Let P be the image of the point Q(7, -2, 5) in the line L: (x - 1)/2 = (y + 1)/3 = z/4, and let R(5, p, q) be a point on L. Then the square…

If the image of the point (4, 4, 3) in the line (x-1)/2 = (y-2)/1 = (z-1)/3 is (a, β, γ), then a + β + γ is equal to:

The square of the distance of the point ( 15/7, 32/7, 7) from the line (x+1)/3 = (y+3)/5 = (z+5)/7 in the direction of the vector i + 4j +…

Let a = i + 2j + k and b = 2i + 7j + 3k. Let L₁: r = (-i + 2j + k) + λ a, λ ∈ ℝ and L₂: r = (j + k) + μ b, μ ∈ ℝ be two lines. If the line…

Let L₁: (x-1)/1 = (y-2)/(-1) = (z-1)/2 and L₂: (x-1)/1 = (y-2)/2 = z/1 be two lines. Let L₃ be a line passing through the point (α, β, γ)…

Let a straight line L pass through the point P(2, -1, 3) and be perpendicular to the lines (x - 1)/2 = (y + 1)/1 = (z - 3)/(-2) and (x -…

Let P be the foot of the perpendicular from the point (1, 2, 2) on the line L: (x-1)/1 = (y + 1)/(-1) = (z - 2)/2. Let the line r = (-i + j…

Let the vertices Q and R of the triangle PQR lie on the line (x+3)/5 = (y-1)/2 = (z+4)/3, QR = 5, and the coordinates of the point P be (0,…

If the image of the point P(1, 0, 3) in the line joining the points A(4, 7, 1) and B(3, 5, 3) is Q(α, β, γ), then α + β + γ is equal to:

Let a line passing through the point (4,1,0) intersect the line L₁: (x - 1)/2 = (y - 2)/3 = (z - 3)/4 at the point A(α, β, γ) and the line…

The distance of the point (7, 10, 11) from the line (x - 4)/1 = (y - 4)/0 = (z - 2)/3 along the line (x - 7)/2 = (y - 10)/3 = (z - 11)/6 is

Let A and B be two distinct points on the line L: (x-6)/3 = (y-7)/2 = (z-7)/(-2). Both A and B are at a distance 2√17 from the foot of…

Let the line L pass through (1, 1, 1) and intersect the lines (x - 1)/2 = (y + 1)/3 = (z - 1)/4 and (x - 3)/1 = (y - 4)/2 = z/1. Then,…

If the equation of the line passing through the point ( 0, -1/2, 0) and perpendicular to the lines r₁ = λ ( i + a j + b k) and r₂ = ( i - j…

Consider the lines L₁: x - 1 = y - 2 = z and L₂: x - 2 = y = z - 1. Let the feet of the perpendiculars from the point P(5, 1, -3) on the…

The distance of the point (7,-2,11) from the line (x-6)/1=(y-4)/0=(z-8)/3 along the line (x-5)/2=(y-1)/(-3)=(z-5)/6 is:

The lines (x - 2)/2 = y/(-2) = (z - 7)/16 and (x + 3)/4 = (y + 2)/3 = (z + 2)/1 intersect at the point P. If the distance of P from the…

Let PQR be a triangle with R(-1, 4, 2). Suppose M(2, 1, 2) is the midpoint of PQ. The distance of the centroid of △ PQR from the point of…

A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x + 2 = 2y = 2z respectively at the points P and Q. If the length of…

Let (α, β, γ) be the foot of the perpendicular from the point (1,2,3) on the line (x+3)/5 = (y - 1)/2 = (z + 4)/3. Then 19(α + β + γ) is…

Let L₁: r = (i - j + 2k) + λ(i - j + 2k), λ ∈ ℝ, L₂: r = (j - k) + μ(3i + j + pk), μ ∈ ℝ, and L₃: r = δ(i + mj - k), δ ∈ ℝ be three lines…

Let a line passing through the point (-1, 2, 3) intersect the lines L₁: (x-1)/3 = (y-2)/2 = (z+1)/(-2) and L₂: (x+2)/(-3) = (y-2)/(-2) =…

The distance of the point Q(0, 2, -2) from the line passing through the point P(5, -4, 3) and perpendicular to the lines r = (-3i + 2k) +…

Let Q and R be the feet of perpendiculars from the point P(a, a, a) on the lines x = y, z = 1 and x = -y, z = -1 respectively. If ∠ QPR is…

Let (α, β, γ) be the mirror image of (2, 3, 5) in the line (x-1)/2 = (y-2)/3 = (z-3)/4. Then 2α + 3β + 4γ is:

Let P and Q be the points on the line (x+3)/8 = (y-4)/2 = (z+1)/2 which are at a distance of 6 units from the point R(1, 2, 3). If the…

The square of the distance of the image of the point (6, 1, 5) in the line (x-1)/3 = y/2 = (z-2)/4, from the origin is:

Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point (11/3, 11/3, 19/3) from the line L along…

Consider the lines L₁ and L₂ given by L₁: (x-1)/2 = (y-3)/1 = (z-2)/2, L₂: (x-2)/1 = (y-2)/2 = (z-3)/3. A line L₃ having direction ratios…

The distance of the point P(4, 6, -2) from the line passing through the point (-3, 2, 3) and parallel to a line with direction ratios 3, 3,…

If the lines (x-1)/1 = (y-2)/2 = (z+3)/1 and (x-a)/2 = (y+2)/3 = (z-3)/1 intersect at the point P, then the distance of the point P from…

Let a line L pass through the point P(2, 3, 1) and be parallel to the line x + 3y - 2z - 2 = 0 i.e. x - y + 2z = 0. If the distance of L…

Let the line L: (x-1)/2 = (y+1)/(-1) = (z-3)/1 intersect the plane 2x + y + 3z = 16 at the point P. Let the point Q be the foot of…

The point of intersection C of the plane 8x + y + 2z = 0 and the line joining the points A(-3, -6, 1) and B(2, 4, -3) divides the line…

If the lines (x-1)/2 = (2-y)/(-3) = (z-3)/α and (x-4)/5 = (y-1)/2 = z/β intersect, then the magnitude of the minimum value of 8αβ is:

Let a line l pass through the origin and be perpendicular to the lines l₁: r = ( i - 11j - 7k) + λ ( i + 2j + 3k), λ ∈ ℝ l₂: r = ( -i + k)…

Let the lines l₁: (x + 5)/3 = (y + 4)/1 = (z - α)/(-2) and l₂: 3x + 2y + z - 2 = 0, x - 3y + 2z - 13 = 0 be coplanar. If the point P(a, b,…