Equation of Line in 3D: JEE Questions & Solutions

Let (,,) 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)+ (2 i+3 j- k). Then…

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

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 - 63 = y - 72 = 7 - z2 is Q(5, b, c), then a^2 + b^2 + c^2 is equal to

Let L be the line x+12 = y+13 = z+36 and let S be the set of all points (a,b,c) on L, whose distance from the line x+12 = y+13 = z-90 along…

The vertices B and C of a triangle ABC lie on the line x1 = 1-y2 = z-23. The coordinates of A and B are (1, 6, 3) and (4, 9, 6)…

If the image of the point P(a, 2, a) in the line x2 = y+a1 = z1 is Q and the image of Q in the line x-2b2 = y-a1 = z+2b-5 is P, then a + b…

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

Let a line L passing through the point P(1,1,1) be perpendicular to the lines x-44= y-11= z-11 and x-171= y-711= z0. Let the line L…

If the distances of the point (1,2,a) from the line x-11= y2= z-11 along the lines L1: x-13= y-24= z-ab and L2: x-11= y-24= z-ac are equal,…

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

Let Q(a,b,c) be the image of the point P(3,2,1) in the line x-11= y-22= z-11. The distance of Q from the line x-93= y-92= z-5-2 is:

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

Let L1: x - 13 = y - 1-1 = z + 10 and L2: x - 22 = y0 = z + 4, where R, be two lines which intersect at the point B. If P is the foot of…

The perpendicular distance of the line x - 12 = y + 2-1 = z + 32 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 3 i + 2 j + 2 k. If the distance of the point…

Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line: x-37 = y-2-1 = z+1-2 Then the area of the right-angled…

The distance of the line x - 22 = y - 63 = z - 34 from the point (1, 4, 0) along the line x1 = y - 22 = z + 33 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 - 63 = y - 72 = z - 7-2…

Let the line passing through the points (-1, 2, 1) and parallel to the line x - 12 = y + 13 = z4 intersect the line x + 23 = y - 32 = z -…

Let P be the image of the point Q(7, -2, 5) in the line L: x - 12 = y + 13 = z4, and let R(5, p, q) be a point on L. Then the square of the…

If the image of the point (4, 4, 3) in the line x-12 = y-21 = z-13 is (a,,), then a + + is equal to:

The square of the distance of the point ( 157, 327, 7) from the line x+13 = y+35 = z+57 in the direction of the vector i + 4 j + 7 k is:

Let a = i + 2j + k and b = 2i + 7j + 3k. Let L1: r = (-i + 2j + k) + a, R and L2: r = (j + k) + b, R be two lines. If the line L3 passes…

Let L1: x-11 = y-2-1 = z-1-1 and L2: x+11 = y-22 = z-22 be two lines. Let L3 be a line passing through the point (,,) and be perpendicular…

Let a straight line L pass through the point P(2, -1, 3) and be perpendicular to the lines x - 12 = y + 11 = z - 3-2 and x - 31 = y - 23 =…

Let P be the foot of the perpendicular from the point (1, 2, 2) on the line x-11 = y + 1-1 = z - 22 Let the line r = (- i + j - 2 k) + ( i…

Let the vertices Q and R of the triangle PQR lie on the line x+35 = y-12 = z+43, QR = 5, and the coordinates of the point P be (0, 2, 3).…

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(,,), then + + is equal to:

Let a line passing through the point (4,1,0) intersect the line L1: x - 12 = y - 23 = z - 34 at the point A(,,) and the line L2: x - 6 = y…

The distance of the point (7, 10, 11) from the line x - 41 = y - 40 = z - 23 along the line x - 72 = y - 103 = z - 116 is

Let A and B be two distinct points on the line L: x-63 = y-72 = z-7-2. Both A and B are at a distance 2 17 from the foot of perpendicular…

Let the line L pass through (1, 1, 1) and intersect the lines x - 12 = y + 13 = z - 14 and x - 31 = y - 42 = z1. Then, which of the…

If the equation of the line passing through the point ( 0, - 12, 0) and perpendicular to the lines r1 = ( 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-61= y-40= z-83 is:

The lines x - 21 = y - 1-1 = z - 78 and x + 34 = y + 23 = z + 21 intersect at the point P. If the distance of P from the line x + 12 = y -…

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 (,,) be the foot of the perpendicular from the point (1,2,3) on the line x5 = y - 12 = z + 43. Then 19( + +) is equal to:

Let L1: r = ( i - j + 2 k) + ( i - j + 2 k), R, L2: r = ( j - k) + (3 i + j + p k), R, and L3: r = ( i + m j - k), R be three lines such…

Let a line passing through the point (-1, 2, 3) intersect the lines L1: x-13 = y-22 = z+1-2 and L2: x+2-3 = y-24 = z-1-2 at points M(,,)…

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 = (-3 i + 2 k) +…

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 a…

Let (,,) be the mirror image of (2, 3, 5) in the line x-12 = y-23 = z-34. Then 2 + 3 + 4 is:

Let P and Q be the points on the line x+38 = y-42 = z+12 which are at a distance of 6 units from the point R(1, 2, 3). If the centroid of…

The square of the distance of the image of the point (6, 1, 5) in the line x-13 = y2 = z-24, from the origin is:

Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point ( 113, 113, 193) from the line L along…

Consider the lines L1 and L2 given by L1: x-12 = y-32 = z-22, L2: x-21 = y-22 = z-33. A line L3 having direction ratios 1, -1, -2…

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-11 = y-22 = z+31 and x-a2 = y+23 = z-31 intersect at the point P, then the distance of the point P from the plane z = a is:

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-12 = y+1-1 = z-31 intersect the plane 2x + y + 3z = 16 at the point P. Let the point Q be the foot of perpendicular from…

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-12 = y-3-3 = z-3 and x-45 = y-12 = z intersect, then the magnitude of the minimum value of 8 is:

Let a line l pass through the origin and be perpendicular to the lines l1: r1 = i + j + 7k + (i + 2j + 3k), R l2: r2 = -i + j + 2k + (i +…

Let the lines l1: x + 53 = y + 41 = z - -2 and l2: 3x + 2y + z - 2 = 0, x - 3y + 2z - 13 = 0 be coplanar. If the point P(a, b, c) on l1 is…