Q. If the real part of $\frac{\overline{\mathcal{z}} + 2}{\overline{\mathcal{z}} - 1}$ is 4, then the locus of the point representing $\mathcal{z}$ in the complex plane is
A) a circle
B) a parabola
C) a hyperbola
D) an ellipse
Explanation
(a)
Let $\mathcal{z =}x + iy$
$\therefore\ Re\ \left( \frac{x - iy + 2}{x - iy - 1} \right) = 4\ \ \ (given)$
$\therefore\ Re\ \left\lbrack \frac{(x + 2) - iy}{(x - 1) - iy} \times \frac{(x - 1) + iy}{(x - 1) + iy} \right\rbrack = 4$
$\Rightarrow \ (x + 2)(x - 1) + y^{2} = 4\left\lbrack (x - 1)^{2} + y^{2} \right\rbrack$
$\Rightarrow \ \ x^{2} + y^{2} - 3x + 2 = 0,$ which represents a circle
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.