UPSC NDA Math Model Paper 3

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.