Andhra Pradesh Board Solutions for Exercise 4: Exercise 1(d)

Show that the points in the Argand diagram represented by the complex numbers $2+2i, -2-2i, -2\sqrt{3}+2\sqrt{3}i$ are the vertices of an equilateral triangle.

Find the eccentricity of the ellipse whose equation is $|z-4|+\left|z-\frac{12}{5}\right|=10$.

If $\frac{z_3-z_1}{z_2-z_1}$ is a real number, show that the points represented by the complex numbers $z_1, z_2, z_3$ are collinear.

Show that the four points in the Argand plane represented by the complex numbers $2+i, 4+3i$, $2+5i, 3i$ are the vertices of a square.

Show that the points in the Argand plane represented by the complex numbers $-2+7i, \frac{-3}{2}+\frac{1}{2}i, 4-3i, \frac{7}{2}\left(1+i\right)$ are the vertices of a rhombus.

Show that the points in the Argand diagram represented by the complex numbers $z_1, z_2, z_3$ are collinear if and only if there exist three real numbers $p, q, r$ not all zero, satisfying $p{z}_1+q{z}_2+r{z}_3=0$ and $p+q+r=0$.

The points $P, Q$ denote the complex numbers $z_1, z_2$ in the Argand diagram. $O$ is the origin. If $z_1{\overline{z}}_2+{\overline{z}}_1{z}_2=0$, then show that $\angle POQ=90°$.

The complex number $z$ has argument $\theta , 0<\theta <\frac{\pi }{2}$ and satisfy the equation $|z-3i|=3$. Then prove that $\left(\cot \theta -\frac{6}{z}\right)=i$.