Question

The equation of the circumcircle of an equilateral triangle is ${x^2} + {y^2} + 2gx + 2fy + c = 0$ and one vertex of the triangle is (1, 1). The equation of incircle of the triangle is
${\text{A}}{\text{. 4(}}{{\text{x}}^2} + {y^2}) = {g^2} + {f^2}$
${\text{B}}{\text{. 4(}}{{\text{x}}^2} + {y^2}) + 8gx + 8fy = (1 + g)(1 - 3g) + (1 + f)(1 - 3f)$
${\text{C}}{\text{. 4(}}{{\text{x}}^2} + {y^2}) + 8gx + 8fy = {g^2} + {f^2}$
\[{\text{D}}{\text{.}}\] None of these

Answer 1

Hint: Centre of the circumcircle is given by, C (-g, -f) and circum-radius is \[R = \sqrt {{{( - g - 1)}^2} + {{( - f - 1)}^2}} \]. Use these formulas to solve the question.

Complete step-by-step answer:
We have been given in the question, the equation of the circumcircle of an equilateral triangle is ${x^2} + {y^2} + 2gx + 2fy + c = 0$ and one vertex of triangle is (1, 1).
Therefore, we know, centre of the circumcircle is given by, C (-g, -f) and circum-radius is \[R = \sqrt {{{( - g - 1)}^2} + {{( - f - 1)}^2}} \].
Since, one vertex of the triangle is given to be (1, 1).
Now, we know that in equilateral the inradius is $r = \dfrac{R}{2}$.
And the centres of both the circles coincide.
Hence, the equation of the incircle is given by-
${(x + g)^2} + {(y + f)^2} = {r^2} = \dfrac{{{R^2}}}{4} = \dfrac{1}{4}[{(g + 1)^2} + {(f + 1)^2}]$
Solving the equation further, we get-
$
  {(x + g)^2} + {(y + f)^2} = {r^2} = \dfrac{{{R^2}}}{4} = \dfrac{1}{4}[{(g + 1)^2} + {(f + 1)^2}] \\
   \Rightarrow 4{(x + y)^2} + 8gx + 8fy = (1 + 2g - 3{g^2}) + (1 + 2f - 3{f^2}) \\
   \Rightarrow 4{(x + y)^2} + 8gx + 8fy = (1 + g)(1 - 3g) + (1 + f)(1 - 3f) \\
 $
Therefore, the equation of the incircle of the given equilateral triangle is $4{(x + y)^2} + 8gx + 8fy = (1 + g)(1 - 3g) + (1 + f)(1 - 3f)$
Hence, the correct option is B.

Note- Whenever such types of questions appear, always note down the equations and the values given in the question to solve easily. Use the given equations and standard results to solve the question. As mentioned in the question the equation of the incircle is given by ${(x + g)^2} + {(y + f)^2} = {r^2}$, substitute the value of r as ${\dfrac{R}{2}}$ and find the equation.