(i) If $\displaystyle A(z_{1}),B(z_{2})$ are two complex numbers, then distance between $\displaystyle z_{1}$ & $\displaystyle z_{2}$ is $\displaystyle AB=\left | z_{1}-z_{2}\right|$
(ii) Line segment joining the points $\displaystyle A(z_{1}), B(z_{2})$ is divided by the point $P(z)$ in the ratio $m : n$, then $\displaystyle z=\frac{mz_{2}+nz_{1}}{m+n}$ where $m,n$ are real numbers.
(iii) Points $\displaystyle z_{1},z_{2},z_{3}$ are collinear if $\displaystyle \begin{vmatrix}z_{1} &\bar{z_{1}} &1 \\z_{2} &\bar{z_{2}} &1 \\z_{3} &\bar{z_{3}} &1 \end{vmatrix}=0$ and the general equation of line joining $\displaystyle A(z_{1}),B(z_{2})$ in non-parametric given by $\displaystyle \begin{vmatrix}z &\bar{z} &1 \\z_{1} &\bar{z_{1}} &1 \\z_{2} &\bar{z_{2}} &1 \end{vmatrix}=0$ and general equation of straight line is $\displaystyle a\bar{z}+\bar{a}z+b=0$ where $a$ is a complex number & $b$ is real, the real slope of the line is $\displaystyle \frac{a+\bar{a}} {i(\bar{a} -a)}$ or equal to $\displaystyle \frac{-Re(a)}{Img(a)}$ and the complex slope of the line is $\displaystyle -\frac{a}{\bar{a}}$ or $\displaystyle \frac{\text{coefficient of }\bar{z}}{\text{coefficient of }z}$.The lines $\displaystyle \bar{a} z+a\bar{z}+\lambda =0$ and $\displaystyle z\bar{a}-\bar{z}a+i\lambda=0$ are respectively parallel and $\displaystyle \perp$ to the line $\displaystyle a\bar{z}+\bar{a} z+b=0$
(iv) The length of $\displaystyle \perp$ from a point $\displaystyle z_{1}$ to the line $\displaystyle \bar{a}z+a\bar{z}+b=0$ is given by $\displaystyle \frac{\left | a\bar{z_{1}}+\bar{a}z_{1}+b \right |}{2\left | a \right |}$.
(v) If the point $\displaystyle A(z_{1}),B(z_{2})$ are to be considered as foci and length of major axis is $2a$ then equation of ellipse is given by  $\displaystyle \left | z-z_{1}\right|+\left| z-z_{2}\right|=2a \implies 2a > \left| z_{1}-z_{2}\right |$.
If major axis is to be considered as transversal axis with length $2a$ such that $\displaystyle 2a<\left|z_{1}-z_{2}\right|$   The equation of hyperbola is given by $\displaystyle \left \| z-z_{1}\mid -\mid z-z_{2} \right \|= 2a$

(vi) If $\displaystyle \left|z-\alpha\right|=r$ $\displaystyle (\alpha$ is complex) is a circle at $\displaystyle \alpha = (p,q)$ & radius $\displaystyle =r$