The ratio in which the join of the points A(2, 1, 5) and B(3, 4, 3) is divided by the plane 2x + 2y - 2z =1, is

5 : 7

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 λ : 1. Then, the coordinates of C are

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

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

$∴ 2 \left(\frac{3λ+2}{λ+1}\right) + 2\left(\frac{4λ+1}{λ+1}\right) - 2\left(\frac{3λ+5}{λ+1}\right)= 1$

$⇒ 8λ - 4 = λ + 1 ⇒λ = \frac{5}{7}$

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