It is given that the angle between the lines whose direction ratios are ⟨2, − 1, 2⟩ And ⟨x, 3, 5 ⟩ is
π
4
. Therefore from the formula for the angle between the lines we get: cos
π
4
=
2x−3+10
√4+1+4√x2+9+25
1
√2
=
2x+7
3√x2+34
1
2
=
(2x+7)2
9(x2+34)
.......squaring both sides Now simplify the above equation to form a quadratic equation as follows: 9x2+306=8x2+56x+98 x2−56x+208=0 (x−52)(x−4)=0 x=52,4 Therefore, the minimum value is 4.