IIT-JEE 1989Question 1
If $a,b,c$ are the numbers between $0$ and $1$ such that the points $z_1=a+i$, $z_2=1+bi$ and $z_3=0$ form an equilateral triangle, then $a=$ ______ and $b=$ ______.
Complex Numbers
Answer$a=2-\sqrt3,\ b=2-\sqrt3$
Solution
Equal sides: $a^2+1=1+b^2$ gives $a=b$ (both positive); $a^2+1=(a-1)^2+(1-a)^2$ gives $a^2-4a+1=0$, so $a=2-\sqrt3$ (as $0<a<1$). Hence $a=b=2-\sqrt3$.