Question

If a and b are real number between 0 and 1 such that the points $z_1=a+i,z_2=b+iand{z}_3=0$
form an equilateral triangle, then

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

Since the triangle with verticals $z_1=a+i,z_2=1+biand{z}_3=0$ is equilateral,we have
$z_1^2+z_2^2+z_3^2=z_1{z}_2+z_2{z}_3+z_3{z}_1$
$\Rightarrow (a+i{)}^2+(1+ib{)}^2+0=(a+i)(i+ib)+0+0$
$\Rightarrow a^2-b^2+2i(a+b)=a-b+i(i+ab)$
Equating real and imaginary parts,
$a^2-b^2=a-b.......\left(i\right)And2(a+b)=1+ab......\left(ii\right)$
From (i), $(a-b)\left[\right(a+b)-1]=0$
$\Rightarrow$ Either $a=bora+b=1$
Taking a=b, we get from (ii)
$4a=1+a^{2}or{a}^2-4a+1=0$
$\therefore a=\frac{4\pm \sqrt{16-4}}{2}=2\pm \sqrt{3}$
Since $0<a<1and0<b<1$,we have $a=b=2-\sqrt{3}$
Taking $a+b=1$ or $b=1-a$,we get from (ii)
$2=1+a(1-a)Or{a}^2-a+1=0$ which gives imaginary values of a.Hence $a=b=2-\sqrt{3}$ is
the required value of a and b.