Question

Find the ratio in which the line joining 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 $\lambda :1$. Then the coordinates of C are

$(\frac{3\lambda +2}{\lambda +1},\frac{4\lambda +1}{\lambda +1},\frac{3\lambda +5}{\lambda +1})$ ...(i)

Since point C lies on the plane 2x+2y-2z=1, therefore coordinates of C must satisfy the equation of the plane

i.e., $2\left(\frac{3\lambda +2}{\lambda +1}\right)+2\left(\frac{4\lambda +1}{\lambda +1}\right)-2\left(\frac{3\lambda +5}{\lambda +1}\right)=1$

$\Rightarrow 8\lambda -4=\lambda +1$

$\Rightarrow \lambda =\frac{5}{7}$

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

Putting $\lambda =\frac{5}{7}$ in (i), the coordinates of the point of division C are $(\frac{29}{12},\frac{9}{4},\frac{25}{6}).$