The combined equation of the lines `L_(1)` and `L_(2)` is `2x^(2)+6xy+y^(2)=0` and that lines `L_(3)` and `L_(4)` is `4x^(2)+18xy+y^(2)=0`. If the angle between `L_(1)` and `L_(4)` be `alpha`, then the angle between `L_(2)` and `L_(3)` will be

Options

A. `(pi)/(2)-alpha`
B. `2alpha`
C. `(pi)/(4)+alpha`
D. `alpha`

Answer

D

Explanation

We observe that the combined equation of the bisectors of the angles between the lines in the first pair is
`(x^(2)-y^(2))/(2-1)=(xy)/(3)rArr 3x^(2)-xy-3y^(2)=0" …(i)"`
and that of the second pair is
`(x^(2)-y^(2))/(4-1)=(xy)/(6)rArr 3x^(2)-xy-3y^(2)=0" ...(ii)"`
Clearly equations (i) and (ii) are same. Thus, the two pairs of lines have the same bisector. Consequently, they are equally inclined to each other. Hence, the angle between `L_(2)` and `L_(3)` is also `alpha`.