Question

Find the ratio in which join the $A(2,1,5)$ and $B(3,4,3)$ is divided by the plane $2x+2y-2z=1$. Also find the coordinates of the point of division.

Answer 1

Let, $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ be two points. Let, $R$ be a point on the line segment joining $P$ and $Q$ internally in the ratio $k:1$. Then co-ordinates of $R$ are,

$R=(\frac{k{x}_2+x_1}{k+1},\frac{k{y}_2+y_1}{k+1},\frac{k{z}_2+z_1}{k+1})$

$A(2,1,5)$ and $B(3,4,3)$

Let, $C$ be the point that divides in the ratio $k:1$ then,

$C=(\frac{3k+2}{k+1},\frac{4k+1}{k+1},\frac{3k+5}{k+1})$

$C$ lies on the plane $2x+2y-2z=1$

$⟹2(\frac{3k+2}{k+1}+\frac{4k+1}{k+1}-\frac{3k+5}{k+1})=1$

$⟹2(3k+2+4k+1-3k-5)=k+1$

$⟹8k-4=k+1$

$⟹7k=5$

$\therefore k=\frac{5}{7}$

i.e., Required ratio $=5:7$

The co-ordinates of the point of division are,

$C=(\frac{3k+2}{k+1},\frac{4k+1}{k+1},\frac{3k+5}{k+1})$

$C=(\frac{15+14}{5+7},\frac{20+7}{5+7},\frac{15+35}{5+7})$

$C=(\frac{29}{12},\frac{27}{12},\frac{50}{12})$