Question

Show that the plane $ax+by+cz+d=0$divides the line joining the points

$(x_1,y_1,z_1)and(x^2,y^2,z^2)$ in the ratio

$-\frac{a{x}_1+b{y}_1+c{z}_1+d}{a{x}_2+b{y}_2+c{z}_2+d}$.

Answer 1

Assume ratio is $\lambda$ : 1

Plane is $ax+by+cz+d=0$

point $(x_1,y_1,z_1)and(x^2,y^2,z^2)$

Assume point of intersection of line and plane is D

D = $(\frac{\lambda x_2+x_1}{\lambda +1},\frac{\lambda y_2+y_1}{\lambda +1},\frac{\lambda z_2+z_1}{\lambda +1})$

As D lies on plane, substitute D in plane equation, we get

$\lambda (a{x}_2+b{y}_2+c{z}_2+d)+a{x}_1+b{y}_1+c{z}_1+d=0$

$\Rightarrow \lambda =\frac{a{x}_1+b{y}_1+c{z}_1+d}{a{x}_2+b{y}_2+c{z}_2+d}$