Show that the condition that the circle `x^2+y^2+2g_1x+2f_1y+c_1=0` bisects the circumference of the circle `x^2+y^2+2g_2x+2f_2y+c_2=0` is `2(g_1-g_2)g_2+2(f_1-f_2)f_2=c_1-c_2`

The circle x^(2) + y^(2) + 2g x + 2fy + c = 0 does not intersect the y-axis if

If the circle x^(2) +y^(2) + 2gx + 2fy + c = 0 bisects the circumference of the circle x^(2) +y^(2) + 2g'x + 2f'y + c' = 0 , them the length of the common chord of these two circles is

If the circle x^2+y^2+2gx+2fy+c=0 bisects the circumference of the circle x^2+y^2+2g^(prime)x+2f^(prime)y+c^(prime)=0 then prove that 2g^(prime)(g-g^(prime))+2f^(prime)(f-f^(prime))=c-c '

If the circle x^2+y^2+2gx+2fy+c=0 bisects the circumference of the circle x^2+y^2+2g^(prime)x+2f^(prime)y+c^(prime)=0 then prove that 2g^(prime)(g-g^(prime))+2f^(prime)(f-f^(prime))=c-c '

Show that the circle passing through the origin and cutting the circles x^2+y^2-2a_1x-2b_1y+c_1=0 and x^2+y^2-2a_2x -2b_2y+c_2=0 orthogonally is |(x^2+y^2,x,y),(c_1,a_1,b_1),(c_2,a_2,b_2)|=0

If circles x^(2) + y^(2) + 2g_(1)x + 2f_(1)y = 0 and x^(2) + y^(2) + 2g_(2)x + 2f_(2)y = 0 touch each other, then

The squared length of the intercept made by the line x=h on the pair of tangents drawn from the origin to the circle x^2+y^2+2gx+2fy+c=0 is (4c h^2)/((g^2-c)^2)(g^2+f^2-c) (4c h^2)/((f^2-c)^2)(g^2+f^2-c) (4c h^2)/((f^2-f^2)^2)(g^2+f^2-c) (d) none of these

If the radical axis of the circles x^2+y^2+2gx+2fy+c=0 and 2x^2+2y^2+3x+8y+2c=0 touches the circle x^2+y^2+2x-2y+1=0 , show that either g=3/4 or f=2

Prove that the general equation of circles cutting the circles x^2 + y^2 + 2g_r x + 2f_r y+c_r = 0, r= 1, 2 orthogonally is : |(x^2+y^2,-x,-y),(c_1,g_1,f_1),(c_2,g_2,f_2)|+k|(-x, -y, 1), (g_1, f_1, 1), (g_2, f_2, 1)| =0

If PQ, PR are tangents from a point P(x_(1),y_(1)) to the circle x^(2)+y^(2)+2gx+2fy+c=0 show that the circumcircle of the triangle PQR is (x-x_(1))(x+g)+(y-y_(1))(y+f)=0