If the direction cosines  l,m,n of two perpendicular lines are connected by the relations λl+3m−5n=0  and  10l2+m2−5n2=0 then λ5 is

The direction cosines of two lines are related by relations al+bm+cn=0  and ul2+vm2+wn2=0 . If these two lines are perpendicular then the condition is ∑a2v+w=0Herea=λ,b=3,c=−5,u=10,v=1,w=−5Hence, ∑a2v+w=0λ2−4+95+2511=0−4λ2+320=0Therefore,  λ=±45⇒λ5=±4