Section Formula Using Complex Numbers

Q.

If three complex numbers are in A.P., then they lie on

1. A circle in the complex plane

2. A straight line in the complex plane

3. A parabola in the complex plane

4. None of these

Q.

If $z$ is a complex number, then the locus of the point $z$ satisfying $\arg \left(\frac{z-i}{z+i}\right)=\frac{\mathrm{\pi }}{4}$ is a

1. circle with centre $\left(-1,0\right)$ and radius $\sqrt{2}$

2. circle with centre $\left(0,0\right)$ and radius $\sqrt{2}$

3. circle with centre $\left(0,1\right)$ and radius $\sqrt{2}$

4. circle with centre $\left(1,1\right)$ and radius $\sqrt{2}$

Q. Let $z_1$ and $z_2$ be two distinct complex numbers and let $z=(1-t)z_1+t{z}_2$ for some real number $t$ with $0<t<1$. If arg$\left(\omega \right)$ denotes the principal argument of a non-zero complex number $\omega$, then

1. $|z-z_1|+|z+z_2|=|z_1-z_2|$
2. arg$(z-z_1)$ = arg$(z-z_2)$
3. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$
4. arg$(z-z_1)$ = arg$(z_2-z_1)$

Q. Let $z_1,z_2,z_3$ be three complex numbers and $a,b,c$ be real numbers not all zero, such that $a+b+c=0$ and $a{z}_1+b{z}_2+c{z}_3=0$, then

1. $z_1,z_2,z_3$ are vertices of a triangle.
2. $z_1,z_2,z_3$ lies on circumference of a circle
3. $z_1,z_2,z_3$ are collinear.
4. None of these.

Q. In $△ABC,A\left(z_1\right),B\left(z_2\right)$ and $C\left(z_3\right)$ are inscribed in the circle $|z|=5$. If $H\left(z_H\right)$ be the orthocentre of $△ABC$, then $z_H=$

1. $\frac{z_1+z_2+z_3}{3}$
2. $z_1+z_2+z_3$
3. $\frac{z_1+z_2+z_3}{2}$
4. $\frac{z_1+2z_2+3z_3}{3}$