Two two straight lines given by `x^2(tan^2theta+cos^2theta)-2xytantheta+y^2sin^2theta=0` make with the axis of x angles such that the difference of their tangents is

Options

A. 4
B. 3
C. 2
D. None of these

Answer

C

Explanation

Let `m_1 and m_2` be the slopes of the lines given by `x^2(tan^2theta+cos^2theta)-2xytantheta +y^2sin^2theta=0`
Then , `m_1+m_2=(2h)/(b)=(2tantheta)/(sin^2theta)=2secthetacosec theta`
and `m_1m_2=(a)/(b)=(tan^2theta+cos^2theta)/(sin^2theta)=sec^2theta+cot^2theta`
Now , `m_1m_2=sqrt((m_1+m_2)^2-4m_1m_2)`
`= sqrt(4sec^2thetacosec^2theta-4(sec^2theta+cot^2theta))`
`=sqrt(4sec^2theta(1+cot^2theta)-4(sec^2theta+cot^2theta))`
`=2sqrt(sec^2thetacot^2theta-cottheta)`
`=2sqrt(cot^2theta(sec^2theta-1))`
`=2sqrt(cot^2thetaxtan^2theta=2)`