If the argument of `(z-a)(barz-b)` is equal to that `((sqrt(3)+i)(1+sqrt(3)i)/(1+i))` where a,b,c are two real number and z is the complex conjugate o the complex number z find the locus of z in the rgand diagram. Find the value of a and b so that locus becomes a circle having its centre at `1/2(3+i)`

Options

A. A. `(3,2)`
B. B. `(2,1)`
C. C. `(2,3)`
D. D. `(2,4)`

Answer

B

Explanation

`(b)` `(z-a)(barz-b)=zbarz-abarz-bz+ab(:'z=x+iy)`
Argument of `(z-a)(barz-b)=((y(a-b))/(x^(2)+y^(2)-ax+bx+ab))`
Also, `((sqrt(3)+i)(1+sqrt(3)i)/(1+i)=(sqrt(3)-sqrt(3)+i(1+3))/(1+i)=2+2i`
Argument of `2+2i=tan^(-1)(2//2)=pi//4`
`impliesx^(2)+y^(2)-(a+b)x-(a-b)y+ab=0`
Thus, the locus of `z` is a circle
Given centre is `(3+i)/(2)`
`implies (a+b)/(2)=(3)/(2)` and `(a-b)/(2)=(1)/(2)`
`impliesa=2`, `b=1`