Foot of perpendicular and image in a line

JEE Main Mathematics · 3D Geometry · 49 questions, latest first

• Let the image of the point P(0,-5,0) in the line (x-1)/2=y/1=(z+1)/-2 be the point R and the image of the point Q(0, -1/2, 0) in…20266 April, Shift 2 · Q24
• Let the image of the point P(1,6, a) in the line L: x/1=(y-1)/2=(z-a+1)/b, b>0, be (a/3, 0, a+c). If S(α, β, γ), α>0, is the…20266 April, Shift 1 · Q15
• If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line x/1=(y-1)/1=(z-2)/2 is 4, then the sum…20265 April, Shift 2 · Q15
• The square of the distance of the point P(5,6,7) from the line (x-2)/2=(y-5)/3=(z-2)/4 is equal to:20265 April, Shift 1 · Q13
• If (2 α+1, α^2-3 α, (α-1)/2) is the image of (α, 2 α, 1) in the line (x-2)/3=(y-1)/2=z/1, then the possible value(s) of α is (are)20264 April, Shift 2 · Q14
• Let the point A be the foot of perpendicular drawn from the point P(a, b, 0) on the line (x-1)/2=(y-2)/1=(z-α)/3. If the midpoint…20262 April, Shift 2 · Q15
• Let a line L passing through the point (1,1,1) be perpendicular to both the vectors 2 i+2 j+ k and i+2 j+2 k. If P(a, b, c) is…20262 April, Shift 1 · Q15
• Let Q(a, b, c) be the image of the point P(3,2,1) in the line (x-1)/1=y/2=(z-1)/1. Then the distance of Q from the line…202628 January, Shift 2 · Q15
• 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-2 b)/2=(y-a)/1=(z+2 b)/-5…202623 January, Shift 2 · Q23
• 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)…202623 January, Shift 1 · Q14
• 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 a^2+b^2+c^2 is equal to202622 January, Shift 1 · Q15
• 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…202621 January, Shift 2 · Q14
• 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+3 j+ k)+λ(2…202621 January, Shift 1 · Q14
• Consider the lines L_1: x-1=y-2=z and L_2: x-2=y=z-1. Let the feet of the perpendiculars from the point P(5,1,-3) on the lines…20257 April, Shift 2 · Q17
• 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…20254 April, Shift 1 · Q15
• 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:20252 April, Shift 2 · Q14
• Let the vertices Q and R of the triangle P Q R lie on the line (x+3)/5=(y-1)/2=(z+4)/3, Q R=5 and the coordinates of the point P…20252 April, Shift 1 · Q13
• 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-2…202529 January, Shift 2 · Q16
• If the image of the point (4,4,3) in the line (x-1)/2=(y-2)/1=(z-1)/3 is (α, β, γ), then α+β+γ is equal to202528 January, Shift 1 · Q15
• 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 R(5, p, q) be a point on L. Then the square of…202524 January, Shift 2 · Q22
• Let in a ABC, the length of the side AC be 6, the vertex B be (1,2,3) and the vertices A, C lie on the line…202524 January, Shift 1 · Q14
• 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…202523 January, Shift 1 · Q15
• The perpendicular distance, of the line (x-1)/2=(y+2)/-1=(z+3)/2 from the point P(2,-10,1), is:202522 January, Shift 2 · Q14
• Let L_1: (x-1)/3=(y-1)/-1=(z+1)/0 and L_2: (x-2)/2=y/0=(z+4)/(α), α ∈ R, be two lines, which intersect at the point B. If P is…202522 January, Shift 1 · Q24
• 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 ____.20249 April, Shift 2 · Q29
• Let P(α, β, γ) be the image of the point Q(1,6,4) in the line x/1=(y-1)/2=(z-2)/3. Then 2 α+β+γ is equal to ____20248 April, Shift 2 · Q29
• Let P(α, β, γ) be the image of the point Q(3,-3,1) in the line (x-0)/1=(y-3)/1=(z-1)/-1 and R be the point (2, 5, -1). If the…20246 April, Shift 2 · Q17
• Let P be the point (10,-2,-1) and Q be the foot of the perpendicular drawn from the point R(1,7,6) on the line passing through…20246 April, Shift 1 · Q28
• Let (α, β, γ) be the image of the point (8,5,7) in the line (x-1)/2=(y+1)/3=(z-2)/5. Then α+β+γ is equal to:20245 April, Shift 2 · Q17
• Let P be the point of intersection of the lines (x-2)/1=(y-4)/5=(z-2)/1 and (x-3)/2=(y-2)/3=(z-3)/2. Then, the shortest distance…20244 April, Shift 2 · Q16
• Consider a line L passing through the points P(1,2,1) and Q(2,1,-1). If the mirror image of the point A(2,2,2) in the line L is…20244 April, Shift 2 · Q29
• If the mirror image of the point P(3,4,9) in the line (x-1)/3=(y+1)/2=(z-2)/1 is (α, β, γ), then 14 (α+β+γ) is:20241 February, Shift 2 · Q15
• Let (α, β, γ) be the mirror image of the point (2,3,5) in the line (x-1)/2=(y-2)/3=(z-3)/4. Then, 2 α+3 β+4 γ is equal to202431 January, Shift 2 · Q16
• The distance of the point Q(0,2,-2) form the line passing through the point P(5,-4, 3) and perpendicular to the lines r=(-3 i+2…202431 January, Shift 1 · Q15
• 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…202431 January, Shift 1 · Q29
• Let (α, β, γ) be the foot of 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 equal to:202430 January, Shift 1 · Q16
• Let the image of the point (1,0,7) in the line x/1=(y-1)/2=(z-2)/3 be the point (α, β, γ). Then which one of the following points…202427 January, Shift 2 · Q16
• 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 line…202427 January, Shift 2 · Q29
• Let N be the foot of perpendicular from the point P(1,-2,3) on the line passing through the points (4,5,8) and (1,-7,5). Then the…202313 April, Shift 2 · Q16
• Let a line l pass through the origin and be perpendicular to the lines l_1: r=( i-11 j-7 k)+λ( i+2 j+3 k), λ ∈ R and l_2: r=(- i+…202311 April, Shift 1 · Q22
• Let P_1 be the plane 3 x-y-7 z=11 and P_2 be the plane passing through the points (2,-1,0),(2,0,-1), and (5,1,1). If the foot of…20238 April, Shift 2 · Q30
• Let λ_1, λ_2 be the values of λ for which the points (5/2, 1, λ) and (-2,0,1) are at equal distance from the plane 2 x+3 y-6…20238 April, Shift 1 · Q28
• The point of intersection C of the plane 8 x+y+2 z=0 and the line joining the points A(-3,-6,1) and B(2,4,-3) divides the line…20231 February, Shift 2 · Q90
• Let the line L: (x-1)/2=(y+1)/-1=(z-3)/1 intersect the plane 2 x+y+3 z=16 at the point P. Let the point Q be the foot of…202331 January, Shift 1 · Q88
• Let a line L pass through the point P(2,3,1) and be parallel to the line x+3 y-2 z-2=0=x-y+2 z. If the distance of L from the…202330 January, Shift 2 · Q89
• Let the co-ordinates of one vertex of A B C be A(0,2, α) and the other two vertices lie on the line (x+α)/5=(y-1)/2=(z+4)/3. For…202329 January, Shift 1 · Q81
• The foot of perpendicular of the point (2,0,5) on the line (x+1)/2=(y-1)/5=(z+1)/-1 is (α, β, γ). Then, which of the following is…202325 January, Shift 2 · Q76
• 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…202325 January, Shift 1 · Q75
• If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3,2,1) and parallel to the planes…202324 January, Shift 2 · Q77