The equation of one of the lines represented by the equation `x^2-2xycottheta-y^2=0`, is

Options

A. `x-y cot theta=0`
B. `x+y tan theta=0`
C. `y sin theta +x(cos theta+1)=0`
D. `x cos theta +y(sintheta+1)=0`

Answer

C

Explanation

Given , `x^2-2xycottheta-y^2=0` .......(i)
it is a homogeneous equation of second degree which represent pair of straight lines passing through origin.
Let `y=mx` be one of the line.
`therefore x^2-2x(mx)cottheta -m^2x^2=0`
`rArr x^2-2x^2mcottheta-m^2x^2=0`
`rArr 1-2mcotehta-m^2=0`
`rArr m^2+2mcot theta-1=0`
`rArr m=(-2cot theta+-sqrt(4cot^2theta+4))/(2)=(-2cot theta+- 2sqrt(cot^2theta+1))/(2)=-cot theta+-cosec theta`
`=(-cot theta cosec theta)or (-cot theta-cosec theta)`
`therefore` Lines are `y=(1-cos theta)/(sin theta)`
`rArr y sin theta-(1-cos theta)x=0`
and `y=(-cot theta- cosec theta)x`
`y=-((1+cos theta))/(sin theta)x`
`rArr y sin theta +(1+cos theta)x=0`