Let the centre of the circle x^2 + y^2 + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x - y = 4 . Let the area of an equilateral triangle inscribed in the circle be 27 3 . Then the square of the length of the chord of the circle on the line x = 1 is _______.

Step-by-step solution

The equation of the circle is x^2 + y^2 + 2gx + 2fy + 25 = 0 . The centre of the circle is (-g, -f) . Since it lies in the first quadrant, -g > 0 and -f > 0 , which implies g The centre lies on the line 2x - y = 4 , so: -2g - (-f) = 4 f = 2g + 4 Let the radius of the circle be R . The side length a of an equilateral triangle inscribed in the circle is a = 3 R . The area of the equilateral triangle is given as 27 3 : 3 4 a^2 = 27 3 3 4 (3R^2) = 27 3 R^2 = 36 The radius of the circle is also given by R^2 = g^2 + f^2