Direction ratios of two lines are 4,k,5 and 2,−1,2. If these lines include an angleπ4, then k can be

If θ is the acute angle between two lines having direction ratios a1,b1,c1and a2,b2,c2then cosθ=a1a2+b1b2+c1c2a12+b12+c12⋅a22+b22+c22It implies that cosπ4=8−k+1016+25+k24+1+412=18−k41+k23Apply square on both sides 12=18−k2941+k2941+k2=2324+k2−36k7k2+369=648−72k7k2+72k−279=0Split the middle term to get the solutions for the equation 7k2+93k−21k−279=0k7k+93−37k+93=0k−37k+93=0Therefore, the possible value of  k is 3