If the plane P passes through the intersection of two mutually perpendicular planes 2x + ky – 5z = 1 and 3kx – ky + z = 5, k < 3 and intercepts a unit length on positive x-axis, then the intercept made by the plane P on the y-axis is

Two given planes mutually perpendicular

2(3k) + k (-k) + (-5) 1 = 0

k = 1, 5

but k < 3      So k = 1

Plane passing through these planes is

$\begin{array}{l}2\mathrm{x}+\mathrm{y}-5\mathrm{z}-1+\mathrm{\lambda }(3\mathrm{x}-\mathrm{y}+\mathrm{z}-5)=0\\ \frac{\mathrm{x}}{\frac{5\mathrm{\lambda }+1}{2+3\mathrm{\lambda }}}+\frac{\mathrm{y}}{\frac{5\mathrm{\lambda }+1}{1-\mathrm{\lambda }}}+\frac{\mathrm{z}}{\frac{5\mathrm{\lambda }+1}{\mathrm{\lambda }-5}}=1\end{array}$

Given $\frac{5\mathrm{\lambda }+1}{2+3\mathrm{\lambda }}=1\Rightarrow \mathrm{\lambda }=\frac{1}{2}$

So intercept on y - axis$=\frac{5\mathrm{\lambda }+1}{1-\mathrm{\lambda }}=7$