Question

If a and b are real 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.
• B.
• C.
• D. None of these

Answer 1

The correct option is B

Since the triangle with vertices $z_1$=a+i,$z_2$=1+bi and $z_3$=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$

$\Rightarrow$ $(a+i{)}^2(1+ib{)}^2+0=(a+i)(1+ib)+0+0$

$\Rightarrow$ $a^2$-$b^2$+2i(a+b)=a-b+i(1+ab)

Equating real and imaginary parts,

$a^2$-$b^2$=a-b ......(i) And

2(a+b)=1+ab

From (i),(a-b)[(a+b)-1]=0

$\Rightarrow$ Either 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<1 and 0<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.