Question

find the direction cosins of the striaght line x=2y=5z

Answer 1

This equation can be written as $\frac{x}{1}=\frac{y}{\frac{1}{2}}=\frac{z}{\frac{1}{5}}$ where $(1, \frac{1}{2}, \frac{1}{5})$ are direction ratios.
Here, $\sqrt{1^{2}+(\frac{1}{2})^{2}+(\frac{1}{5})^{2}}=\frac{\sqrt{129}}{10}$.
Hence, direction cosines are $(\frac{10}{\sqrt{129}},\frac{5}{\sqrt{129}},\frac{2}{\sqrt{129}})$.

Answer 2

take a point P  on the straight line  L  x = 2 y = 5 z.

   Take x = 10  (lcm of 2 and 5)      y = 5 and z = 2.          P = (10, 5 , 2)

   this straight line L  goes through the origin O.
   the distance  OP = r  =  √(10²+5²+2²) = √129

   Direction cosines are   (cos α,  cos β,  cos γ)  = ( x/r , y/r,  z/r )
                    = ( 10/√129,  5/√129, 2/√129)