Show that the two straight lines
`x^2(tan^2theta+cos^2theta)-2xy tantheta+y^2sin^2theta=0`
Make with the axis of x angles such that the difference of their tangents is 2 .

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^(2)theta+cos^(2)theta)-2xy tan theta +y^(2)sin^(2)theta=0`. Then,
`m_(1)+m_(2)=(2tan theta)/(sin^(2)theta)=2sec theta" cosec "theta`
and `m_(1)m_(2)=(tan^(2)theta+cos^(2)theta)/(sin^(2)theta)=sec^(2)theta+cot^(2)theta`
`therefore" "m_(1)-m_(2)=sqrt((m_(1)+m_(2))^(2)-4m_(1)m_(2))`
`rArr" "m_(1)-m_(2)=sqrt(4sec^(2)theta" cosec"^(2)theta-4(sec^(2)theta+cot^(2)theta))`
`rArr" "m_(1)-m_(2)=sqrt(4sec^(2)theta(1+cot^(2)theta)-4(sec^(2)theta+cot^(2)theta))`
`rArr" "m_(1)-m_(2)=2sqrt(sec^(2)thetacot^(2)theta-cot^(2)theta)`
`rArr" "m_(1)-m_(2)=2sqrt(cot^(2)theta(sec^(2)theta-1))=2`