goals

Consider a complex number $$w = \dfrac{z - i}{2z + 1}$$, where $$z = x + iy$$ and $$x, y \epsilon R$$.

(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 $$\displaystyle =r$$

(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 $$\displaystyle =r$$

(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 $$\displaystyle =r$$

(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 $$\displaystyle =r$$

Circum-centre of $$\triangle ABC$$ with vertices $$A(Z_1), B(Z_2)$$ and $$C(z_3)$$ is $$S(z_0)$$.

The altitude from vertex $$A$$ to side $$BC$$ meets the circumcircle at $$E(z_4)$$.

The altitude from vertex $$A$$ to side $$BC$$ meets the circumcircle at $$E(z_4)$$.