Question

Show that the plane `a x+b y+c z+d=0` divides the line joining the points `(x_1, y_1, z_1)a n d(x_2, y_2, z_2)` in the ratio `(a x_1+b y_1+c z_1+d)/(a x_2+b y_2+c z_2+d)` .

Answer 1

`C((lambdax_2)/(lambda+1),(lambday_2+y_1)/(9lambda+1),(lambdaz_2+z_1)/(lambda+1))`
`ax+by+cz+d=0`
`a((lambdax_2+x_1)/(lambda+1))+b((lambday_2+y_1)/(lambda+1))+c((lambdaz_2+z_1)/(lambda+1))+d=0`
`alambdax_2+blambday_2+by_1+cz_2+cz_1+dlambda+d=0`
`lambda(ax_2+by_2+cz_2+d)+(ax_1+by_1+cz_1+d)=0`
`lambda=-(ax_1+by_1+cz_1+d)/(ax_2+by_2+cz_2+d)`.