Question

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio  \(\frac{ax_1+ by_1+cz_1+d}{ax_2+ by_2+cz_2+d}\)

Answer 1

Given: A(x₁, y₁, z₁) and B(x₂, y₂, z₂) 

To prove: the ratio in which the line segment AB is divided by the plane ax + by + cz + d = 0 is \(\frac{ax_1+ by_1+cz_1+d}{ax_2+ by_2+cz_2+d}\)

Formula used: 

Section Formula: 

A line AB is divided by C in m:n where A(x, y, z) and B(a, b, c).

The coordinates of C is given by,

\(\Big(\frac{nx+ma}{m+n},\frac{ny+mb}{m+n},\frac{nz+mc}{m+n}\Big)\)

Let C(x, y, z) be any point on given plane and C divides AB in ratio k: 1

Therefore, m = k and n = 1 

A(x₁, y₁, z₁) and B(x₂, y₂, z₂) 

Coordinates of C using section formula:

Therefore, m = k and n = 1 

A(x₁, y₁, z₁) and B(x₂, y₂, z₂) 

Coordinates of C using section formula:

The plane divides AB in the ratio 

\(\frac{ax_1+ by_1+cz_1+d}{ax_2+ by_2+cz_2+d}\)