The equation of the plane passing through the line intersection of the plane →r.(→i+3→j)−6=0 and →r.(3→i−→j−4→k)=0 is →r[(→i+3→j)+λ(3→i−→j−4→k)]=6+λ(0) [Using →r.[→n1+λ→n2]=d1+λ2]
i.e., →r.[(1+3λ)→i+(3−λ)→j−4λ→k]−6=0 ---- (1)
Now distnace of (1) from (0, 0) is ∣∣∣(o→i+0→j+0→k[((1+3λ)→i+(3−λ)→j−4λ→k]−6)])∣∣∣√(1+3λ)2+(3−λ)2+(−4λ)2=1
(6√(1+3λ)2+(3−λ)2+(−4λ)2)2=1⇒3626λ2+10=1⇒λ=±1
Substituting the value of λ in (1), we get →r.[4→i+2→j−4→k]−6=0
i.e.,→r.[2→i+→j−2→k]=3
and →r.[−2→i+4→j+4→k]−6=0 i.e., →r.[→i−2→j−2→k]+3=0