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The combined equation of the lines L1and...

The combined equation of the lines `L_1`and `L_2` is `2x^2+6xy+y^2=0`, and that of the 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_1 and L_3` will be .

A

`(pi)/(2)-alpha`

B

`2alpha`

C

`(pi)/(4)+alpha`

D

`alpha`

Text Solution

Verified by Experts

The correct Answer is:
D
We observe that the combined equation of the bisector of the angles btween the lines in the first pair is `(x^2-y^2)/(2-1)=(xy)/(3)[because a=2,b=1 and h=3]`
`rArr 3x^2-xy-3y^2=0`.......(i)
and that of the second pair is `(x^2-y^2)/(4-1)=(xy)/(9)[because a=4,b=1 and h=9]`
`rArr 3x^2-xy-3y^2=0` ........(ii)
Clearly, Eqs. (i) and (ii) are same. Thus , the two pairs of lines have the same bisecter. Consequently, they are equally inclined to each other. Hence , theangle between `L_2and L_3` is also `alpha`.
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