Question

(i) If $A\left(z_1\right),B\left(z_2\right)$ are two complex numbers, then distance between $z_1$ & $z_2$ is $AB=|z_1-z_2|$

(ii) Line segment joining the points $A\left(z_1\right),B\left(z_2\right)$ is divided by the point $P\left(z\right)$ in the ratio $m:n$, then $z=\frac{m{z}_2+n{z}_1}{m+n}$ where $m,n$ are real numbers.

(iii) Points $z_1,z_2,z_3$ are collinear if $\left|\begin{array}{ccc}{z}_1& \overline{z_1}& 1\\ z_2& \overline{z_2}& 1\\ z_3& \overline{z_3}& 1\end{array}\right|=0$ and the general equation of line joining $A\left(z_1\right),B\left(z_2\right)$ in non-parametric given by $\left|\begin{array}{ccc}z& \overline{z}& 1\\ z_1& \overline{z_1}& 1\\ z_2& \overline{z_2}& 1\end{array}\right|=0$ and general equation of straight line is $a\overline{z}+\overline{a}z+b=0$ where $a$ is a complex number & $b$ is real, the real slope of the line is $\frac{a+\overline{a}}{i(\overline{a}-a)}$ or equal to $\frac{-Re\left(a\right)}{Img\left(a\right)}$ and the complex slope of the line is $-\frac{a}{\overline{a}}$ or $\frac{\text{coefficient of}\overline{z}}{\text{coefficient of}z}$.The lines $\overline{a}z+a\overline{z}+\lambda =0$ and $z\overline{a}-\overline{z}a+i\lambda =0$ are respectively parallel and $\perp$ to the line $a\overline{z}+\overline{a}z+b=0$

(iv) The length of $\perp$ from a point $z_1$ to the line $\overline{a}z+a\overline{z}+b=0$ is given by $\frac{|a\overline{z_1}+\overline{a}{z}_1+b|}{2\left|a\right|}$.

(v) If the point $A\left(z_1\right),B\left(z_2\right)$ are to be considered as foci and length of major axis is $2a$ then equation of ellipse is given by $|z-z_1|+|z-z_2|=2a⟹2a>|z_1-z_2|$.

If major axis is to be considered as transversal axis with length $2a$ such that $2a<|z_1-z_2|$ The equation of hyperbola is given by $\parallel z-z_1\mid -\mid z-z_2\parallel =2a$

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

Let $\alpha ,\beta$ be two fixed non zero complex numbers & '$z$' a variable complex number such that the lines $\alpha \overline{z}+\overline{\alpha }z+1=0$ and $\beta \overline{z}+\overline{\beta }z+1=0$ are mutually perpendicular lines, then value of $\alpha \overline{\beta }+\beta \overline{\alpha }$ equals

• A. $2$
• B. $-2$
• C. $0$
• D. $1$

Answer 1

Answer: (C)

$0$

Slope of first line $m_1$

$=\frac{-\alpha }{\overline{\alpha }}$

Similarly slope of second-line $m_2$

$=\frac{-\beta }{\overline{\beta }}$

Now

$m_1.m_2=-1$ in co-ordinate plane.

Thus, corresponding

${\theta }_1+{\theta }_2=0$

$\therefore$ $-\frac{\alpha }{\overline{\alpha }}-\frac{\beta }{\overline{\beta }}=0$

Therefore $\alpha \overline{\beta }+\beta \overline{\alpha }=0$