Question

The locus of the point of intersection of lines `sqrt3x-y-4sqrt(3k)`=0 and `sqrt3kx+ky-4sqrt3=0` for different value of k is a hyperbola whose eccentricity is 2.

Answer 1

True
Given equation of line are
`sqrt3x-y-4sqrt3k`=0 …(i)
and `sqrt3kx+ky-4sqrt3=0`
From Eq. (i) `4sqrt3k=sqrt3x-y`
`rArr k=(sqrt3x-y)/(4sqrt3)`put in Eq. (ii), we get
`sqrt3x((sqrt3x-y)/(4sqrt3))+((sqrt3x-y)/(4sqrt3))y-4sqrt3=0`
`rArr1/4(sqrt3x^(2)-xy)+1/4(xy-(y^(2))/(sqrt3))-4sqrt3=0`
`rArr (sqrt3)/4x^(2)-(y^(2))/(4sqrt3)-4sqrt3=0`
`rArr 3x^(2)-y^(2)-48=0`
`rArr 3x^(2)-y^(2)=48`,which is hyperbola.