Circles `x^2+y^2 +2gx + 2fy=0` and `x^2+y^2 +2g'x + 2f'y = 0` touch externally, if

If two cricles x^2 + y^2 + 2gx + 2fy = 0 and x^2 +y^2 + 2g'x + 2f'y = 0 touch each other, then

if the two circles x^2 + y^2 + 2gx + 2fy = 0 and x^2 + y^2 + 2g'x + 2 f'y = 0 touch each other then show that f'g = fg'

If the circles x ^(2) +y^(2) +2gx +2fy =0 and x ^(2) +y^(2) +2g'x+ 2f'y=0 touch each other then-

Show that the two circles x^(2) + y^(2) + 2gx + 2fy = 0 and x^(2) + y^(2) + 2g'x+ 2f'y = 0 will touch each other if f'g = g'f.

If two circle x^(2)+y^(2)+2gx +2fy=0 and x^(2)+y^(2)+2g'x+2f'y=0 touch each other then f'g =fg'.

If the circles x^(2)+y^(2)+2gx+2fy=0 and x^(2)+y^(2)+2g'x+2f'y=0 touch each other, then

If two circles x^2+y^2+2gx+2fy=0 and x^2+y^2+2g'x+2f'y=0 touch each other , then ((f')/(f))(g/(g')) =___

If two circles x^2+y^2+2gx+2fy=0 and x^2+y^2+2g'x+2f'y=0 touch each other , then ((f')/(f))(g/(g')) =___

If two circle x^(2)+y^(2)+2gx +2fy=0 and x^(2)+y^(2)+2g'x+2f'y=0 touch each other then proove that f'g =fg'.