Question

If $a,b$ are the numbers between $0$ and $1$ such that the points $z_1=a+i,z_2=1+bi$ and $z_0=0$ form an equilateral triangle, then

• A. $a=2-\sqrt{3}\text{and}b=2-\sqrt{3}$
• B. $a=2-\sqrt{3}\text{and}b=2+\sqrt{3}$
• C. $a=2+\sqrt{3}\text{and}b=2-\sqrt{3}$
• D. $a=2+\sqrt{3}\text{and}b=2+\sqrt{3}$

Answer 1

The correct option is A $a=2-\sqrt{3}\text{and}b=2-\sqrt{3}$

$\because z_0,z_1,z_2$ forms a equilateral triangle
$\therefore \frac{z_1-z_0}{z_2-z_0}=e^{\pm i\pi /3},\frac{|z_1-z_0|}{|z_2-z_0|}=1\Rightarrow a+i=(1+bi)(\frac{1}{2}\pm i\frac{\sqrt{3}}{2}),a^2+1=1+b^2\Rightarrow a+i=\frac{1\mp \sqrt{3}b}{2}+i\frac{(b\pm \sqrt{3})}{2},a^2=b^2\Rightarrow b=2\mp \sqrt{3},a=2\mp \sqrt{3}$
but $a,b\in (0,1)$
$\therefore b=2-\sqrt{3},a=2-\sqrt{3}$