Two vertices of an equilateral triangle are `(-1,0)` and (1, 0), and its third vertex lies above the x-axis. The equation of its circumcircel is ____________

Options

A. `x^(2)+y^(2)=1`
B. `sqrt(3)(x^(2)+y^(2))+2y-sqrt(3)=0`
C. `sqrt(3)(x^(2)+y^(2))-2y-sqrt(3)=0`
D. none of these

Answer

C

Explanation

The circumcentre and centroid of an equilateral triangle are same. Let G be the centroid.

We have, AB=2
`:. ` OC = Altitude of `DeltaABC= (sqrt(3))/(2)xx2=sqrt(3)`
But, `OG=(1)/(3)OC rArr OG = (1)/(sqrt(3))`
Thus, the coordinates of G are ,`( 1//sqrt(3))`
`:. AG=sqrt(1+(1)/(3))=(2)/(sqrt(3))`
So, the equation of the circumcircle of `DeltaABC` is
`(x-0)^(2)+(y-(1)/(sqrt(3)))^(2)=((2)/(sqrt(3)))^(2)`
`rArr x^(2)+y^(2)-(2y)/(sqrt(3))-1=0 rArr sqrt(3)(x^(2)+y^(2))-2y-sqrt(3)=0`