Let the equation of a straight line `L` in complex form be `abarz+baraz+b=0`, where is a complex number and `b` is a real number then

Options

A. the straight line `(z-c)/a +(i(barz-barc))/a=0` makes a angle of `45^(@)` with `L` and passed through a point `c` (where `c` is a complex number)
B. the straight line `(z-c)/a=(i(barz-barc))/(bara)` makes an angle of `45^(@)` with `L` and passes through `a` and `c` (where `c` is a complex number)
C. the complex slope of the line `L` is `-a/bara`
D. the complex slope of the line `L` is `a/bara`

Answer

A, B, C

Explanation

Let `P(z)` be any point on the required line.
Then, `(vec(CP))/(|vecCP|)` i.e. `(z-c)/(|z-c|)` is a unit vector parallel to it
Let `A(z_(1))` and `B(z_(2))` be two points on
`baraz+abarz+b=0` then `(z_(2)-z_(1))/(|z_(2)-z_(1)|)` is a unit vector parallel to the line
`abarz+baraz+b=0`
`(z-c)/(|z-c|)=(z_(2)-z_(1))/(|z_(2)-z_(1)|)e^(+-i((pi)/4))`
`((z-c)^(2))/((z-c)(barz-barc))=((z_(2)-z_(1))^(2))/((z_(2)-z_(1))(barz_(2)-barz_(1)))e^(+-i(pi)/2)`
`((z-c)/(barz-barc))= +- i ((z_(2)-z_(1))/(barz_(2)-barz_(1)))`.......(1)
`:' A(z_(1)` and `B(z_(2))` are on the line `abarz+baraz+b=0`therefore `abarz_(1)+baraz_(1)+b=0`
`abarz_(2)+baraz+b=0`
`implies-a/a=(z_(2))/(barz_(2))=((z_(2)-z_(1))/(barz_(2)-barz_(1)))`......(2)
From equation (1) and (2) we get `(z-c)/a+-(i(barz-barc))/(bara)=0`