Q.

The equation of the circumcircle of an equilateral triangle  is x2+y2+2gx+2fy+c=0 and one vertex of the triangle is (1, 1). The equation of the incircle of the triangle is

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a

4x2+y2=g2+f2

b

4x2+y2+8gx+8fy=(1g)(1+3g)+(1f)(1+3f)

c

4x2+y2+8gx+8fy=g2+f2

d

none of these

answer is B.

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Detailed Solution

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In an equilateral triangle, circumcenter and incenter are coincident. 

Therefore, Incenter (g,f)

  Since  Point (1,1) lies on the circle. 

12+12+2g+2f+c=0 or  c=2(g+f+1)

  Also, in an equilateral triangle,

 Circumradius =2× Inradius   Inradius =12×g2+f2c

Therefore, the eqlration of the incircle is (x+g)2+(y+f)2=14g2+f2c =14g2+f2+2(g+f+1)4x2+y2+8gx+8fy=1-g1+3g+1-f1+3f

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The equation of the circumcircle of an equilateral triangle  is x2+y2+2gx+2fy+c=0 and one vertex of the triangle is (1, 1). The equation of the incircle of the triangle is