Equation of a plane at a distance $\sqrt {{2 \over {21}}}$ from the origin, which contains the line of intersection of the planes x $-$ y $-$ z $-$ 1 = 0 and 2x + y $-$ 3z + 4 = 0, is :

Required equation of plane<br><br>${P_1} + \lambda {P_2} = 0$<br><br>$(x - y - z - 1) + \lambda (2x + y - 3z + 4) = 0$<br><br>Given that its dist. From origin is ${2 \over {\sqrt {21} }}$<br><br>Thus, $${{|4\lambda - 1|} \over {\sqrt {{{(2\lambda + 1)}^2} + {{(\lambda - 1)}^2} + {{( - 3\lambda - 1)}^2}} }} = {{\sqrt 2 } \over {\sqrt {21} }}$$<br><br>$\Rightarrow 21{(4\lambda - 1)^2} = 2(14{\lambda ^2} + 8\lambda + 3)$<br><br>$\Rightarrow 336{\lambda ^2} - 168\lambda + 21 = 28{\lambda ^2} + 16\lambda + 6$<br><br>$\Rightarrow 308{\lambda ^2} - 184\lambda + 15 = 0$<br><br>$\Rightarrow 308{\lambda ^2} - 154\lambda - 30\lambda + 15 = 0$<br><br>$\Rightarrow (2\lambda - 1)(154\lambda - 15) = 0$<br><br>$\Rightarrow \lambda = {1 \over 2}$ or ${{15} \over {154}}$<br><br>for $\lambda = {1 \over 2}$ reqd. plane is $4x - y - 5z + 2 = 0$