If `(3)/(2+e^(itheta)) =ax + iby`, then the locous of `P(x,y)` will represent

Options

A. ellipse of a =1,b=2
B. circle if a=b=1
C. pair of straight line if a=1,b=0
D. None of these

Answer

A, B, C

Explanation

`(3)/(2+e^(itheta))= ax + iby`
`rArr (3) /(ax + iby) - 2 = e^(itheta)`
`rArr (3-2ax - 2iby)/(ax + iby ) = e^(itheta)`
`|(3-2ax-2iby)/(ax + iby)| = |e^(itheta)|=1`
`rArr (3-2ax)^(2) + 4b^(2)y^(2) = a^(2)x^(2)+ b^(2)y^(2)`
When a =1 and b=2 we have,
`(3-2x)^(2) + 16y^(2) =x^(2) + 4y^(2)`
`rArr 3x^(2) + 12y ^(2) -12y^(2) + 9=0`
`rArr x^(2) - 4x + 4y^(2) + 3 = 0`, which is ellispe
When a = b= 1 , we have
`(3-2x)^(2) + 4y^(2) = x^(2) + y^(2)`
`rArr 3x^(2) + 3y^(2) - 12x + 9 =0`
or `x^(2) + y^(2) - 4x + 3=0` , which is circle ltbegt When a= 1 and b = 0, we have
`93-2x)^(2) = x^(2),` which is pair of straight lines