Equation of Conics in Complex Form

Q. If $z$ is a complex number, not purely real such that imaginary part of $z-1+\frac{1}{z-1}$ is zero, then locus of $z$ is

1. a straight line parallel to $x$-axis
2. a circle of radius $1$ unit
3. a parabola with axis of symmetry parallel to $x$-axis
4. a hyperbola

Q. Let $f\left(x\right)=x^3-3x^2-9x+9,$ then which of the following(s) is(are) correct

1. $f\left(x\right)$ has a local minima at $x=3$
2. $f\left(x\right)$ has a local maxima at $x=-1$
3. $f\left(x\right)$ has a local minima at $x=-1$
4. $f\left(x\right)$ has a local maxima at $x=3$

Q. If $z=x+iy$ and $|z-1{|}^2+|z+1{|}^2=4$, then the locus of $z$ is

1. $x^2-y^2=1$
2. $x^2+y^2=1$
3. $x^2-y^2=4$
4. $x^2+y^2=2$

Q. The locus of point $z$ satisfying $Re\left(\frac{1}{z}\right)=k$, where $k$ is a non-zero real number is

1. a straight line
2. a circle
3. an ellipse
4. a hyperbola

Q. If $z$ is a complex number, not purely real such that imaginary part of $z-1+\frac{1}{z-1}$ is zero, then locus of $z$ is

1. a straight line parallel to $x$-axis
2. a circle of radius $1$ unit
3. a parabola with axis of symmetry parallel to $x$-axis
4. a hyperbola

Q. If $z$ is a complex number that satisfies $z^2+z|z|+|z{|}^2=0,$ then the locus of $z$ is

1. a circle
2. a straight line
3. a pair of straight lines
4. an ellipse