Q.
If an equilateral triangle $A B C$ with vertices at $z_1, z_2$ and $z_3$ be inscribed in a circle $|z|=2$ and again a circle is inscribed in $\triangle ABC$ touching sides $AB , BC$ and $CA$ at $D \left( z _4\right), E \left( z _5\right)$ and $F \left( z _6\right)$ respectively.
Column I
Column II
P
Value of $\operatorname{Re}\left( z _1 \overline{ z }_2+ z _2 \overline{ z }_3+ z _3 \overline{ z }_1\right)$ is equal to
1
2
Q
If $\frac{4 z_1}{z_3}=a(-1+i \sqrt{3})$ then a is equal to
2
-6
R
$\quad\left|z_1+z_2\right|^2+\left|z_2+z_3\right|^2+\left|z_3+z_1\right|^2$ is equal to
3
12
S
If $P$ is any point on incircle, then $DP ^2+ EP ^2+ FP ^2$ is
4
6
| Column I | Column II | ||
|---|---|---|---|
| P | Value of $\operatorname{Re}\left( z _1 \overline{ z }_2+ z _2 \overline{ z }_3+ z _3 \overline{ z }_1\right)$ is equal to | 1 | 2 |
| Q | If $\frac{4 z_1}{z_3}=a(-1+i \sqrt{3})$ then a is equal to | 2 | -6 |
| R | $\quad\left|z_1+z_2\right|^2+\left|z_2+z_3\right|^2+\left|z_3+z_1\right|^2$ is equal to | 3 | 12 |
| S | If $P$ is any point on incircle, then $DP ^2+ EP ^2+ FP ^2$ is | 4 | 6 |
Complex Numbers and Quadratic Equations
Solution:
