Question

Find the ratio in which the join of 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 given plane intersects AB at a point C and let the required ratio be $\lambda :1$.

Then, the coordinates of C are

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

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

$\therefore 2\left(\frac{3\lambda +2}{\lambda +1}\right)+2\left(\frac{4\lambda +1}{\lambda +1}\right)-2\left(\frac{3\lambda +5}{\lambda +1}\right)=1$ or $\lambda =\frac{5}{7}$.

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

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