Find the equation of the plane passing through the line of intersection of the planes 𝑟.(𝑖+3𝑗)=0 and 𝑟.(3𝑖−𝑗−𝑘)=0 whose perpendicular distance from origin is unity.

The equation of the plane passing through the line intersection of the plane 𝑟 (𝑖+3𝑗)=0 and 𝑟 (3𝑖−𝑗−𝑘)=0 is 𝑟[(𝑖+3𝑗)+λ(3𝑖−𝑗−𝑘)]=0 [Using 𝑟.[𝑛1+λ𝑛2]=d1+λd2] i.e., 𝑟.[(1+3λ)𝑖+(3−λ)𝑗−λ𝑘]=0 — (1)
Now distance of (1) from (0, 0) is √((1+3λ)²+(3−λ)²+(−λ)²)=1
(6λ²+10)=1
λ=±1
Substituting the value of λ in (1), we get
𝑟.[4𝑖+2𝑗−𝑘]=0 i.e., 𝑟.[2𝑖+𝑗−𝑘]=0
and 𝑟.[−2𝑖+4𝑗+𝑘]=0 i.e., 𝑟.[−2𝑖+4𝑗+𝑘]=0