Question

If A(x₁, y₁, z₁) and B(x₂, y₂, z₂) are two distinct points in space and a point P(x, y, z) divides AB in the ratio m : n internally then Prove that.

x = (mx₂ + nx₁)/(m + m), y = (my₂ + ny₁)/(m + n), z = (mz₂ + nz₁)/(m + n)

Answer 1

Draw AL, PN and BM perpendiculars to the xy-plane. 

Clearly the feet of the perpendiculars L, M, N are collinear and L = (x1,y1), M -(x₂, y₂), N = (x,y). 

Now, draw a line through P which is parallel to the line LM meets AL produced at C and BM at D. 

Clearly the triangle PAC and PDB are equiangular and hence similar.

Since the lines AL, PN and BM are parallel, we have,

Remark: 

(i) It P divides AB externally in the ratio m : n, then the coordinates of the point P are

Observations: 

• The point P divides AB internally in the ratio k : 1 iff k > 0 (i.e., k is + ve) 
•  The point P divides AB externally in the ratio |k| : 1 iff k < 0 (i.e., k is – ve)