Test: Distance Formula 3D Geometry - JEE MCQ

Let the given points be A (1, 2, 3) and B (3, 2, -1) Let P(x, y, z) be any point which is equidistant from the points A and B.
Then PA = PB 


⇒ (x - 1)² + (y - 2)² + (z - 3)²
= (x - 3)² + (y - 2)² + (z +1)²
⇒ x² +1 - 2x + y² + 4 - 4y + z² + 9 - 6z
= x² + 9 - 6x + y² + 4 - 4y + z² +1+2z
⇒ 6x - 2x - 4y + 4y -  6z - 2z = 0
⇒ 4x - 8z = 0
⇒ x - 2z = 0
This is the required equation of the set of points in reference.

Given points (1,2,3)
Pt A (1,0,0) on x axis
Pt B (0,2,0) on y axis
Pt C (0,0,3) on z axis
A² = [0 + 2 + 3]
A² = 13
B² = [1 + 0 + 3]
B² = 10
C² = [1 + 2 + 0]
C² = 5
 
a)2A²C² = 13B²
⇒ 2(13)(5) = 13(10)
⇒ 130 = 130 {true}
 
b) A² = 2C²
⇒ (13) = 2(5)
⇒ 13 = 10 {false}
 
c)B² = 3C²
⇒ 10 = 3(5)
⇒ 10 = 15 {false}
 
d)A² = B² + C²
⇒ (13) = 10 + 5
⇒ 13 = 15 {false}

AB =  [(2−4)² +(3−2)² +(4−8)²]^1/2
 AB=  [(−2)² + (1)² + (−4)²]^1/2
 AB =  (21)^1/2
Similarly you find that BC=  (43)^1/2
CD= (33)^1/2  and DA= (43)^1/2 
Hence opposite sides of quadrilateral are equal, Now we check the diagonals
AC=  [(-1-4)² + (−2-7)² + (1−8)²]^1/2
AC=  (155)^1/2
similarly BD=  (3)^1/2
Diagonals are not equal
direction ratio of line passing through AB is (-2,-4,-4)
direction ratio of line passing through  CD is (2,4,4), As the dr of AB and CD are proportional which means AB is parallel to CD,
Similarly check for BC and DA then you will find that they are also parallel
Hence it is parallelogram.

Let the point on Z axis be given as (0,0,z).  The distance between (1,2,3) and (0,0,z) is given as [(1)² + (2)² + (3-z)²]^½ = (21)^1/2
5+(3−z)²=21
z²−6z−7=0
z=7,z = −1
Hence points are (0,0,7),(0,0,−1)