Question

Find the ratio in which the points join 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

Suppose the plane 2x+2y-2z=1 divides the line joining the points A(2,1,5) and B(3,4,3) at a point C in the ratio k:1.

Then, the coordinates of C are $(\frac{3k+2}{k+1},\frac{4k+1}{k+1},\frac{3k+5}{k+1})$

Since, point C lies on the plane 2x+2y-2z=1.

$\therefore$ Coordinates of C must satisfy the equation of plane, i.e., $2\left(\frac{3k+2}{k+1}\right)+2\left(\frac{4k+1}{k+1}\right)-2\left(\frac{3k+5}{k+1}\right)=1$

$\Rightarrow \frac{6k+4+8k+2-6k-10}{k+1}=1\Rightarrow 8k-4=k+1\Rightarrow k=\frac{5}{7}$

So, the required ratio is $\frac{5}{7}:1$ or 5:7.

Substitute $k=\frac{5}{7}$ is Eq. (i), we get

Coordinates of $C(\frac{3\times \frac{5}{7}+2}{\frac{5}{7}+1},\frac{4\times \frac{5}{7}+1}{\frac{5}{7}+1},\frac{3\times \frac{5}{7}+5}{\frac{5}{7}+1})$ or $(\frac{29}{12},\frac{9}{4},\frac{25}{6})$.