If two circles `x^(2)+y^(2)+2a_(1)x+2b_(1)y=0` and `x^(2)+y^(2)+2a_(2)x+2b_(2)y=0` touches then show that `a_(1)b_(2)=a_(2)b_(1)`

If the pair of lines a_(1)x^(2)+2h_(1)xy+b_(1)y^(2)=0 and a_(2)x^(2)+2h_(2)xy+b_(3)y^(2)=0 have one line in common then show that (a_(1)b_(2)-a_(2)b_(1))^(2)+4(a_(1)h_(2)-a_(2)h_(1))(b_(1)h_(2)-b_(2)h_(1))=0

Show that the circles x^(2) +y^(2) + 2ax + c=0 and x ^(2) + y^(2) + 2by + c=0 to touch each other if (1)/(a^(2)) + (1)/( b^(2)) = (1)/( c )

The point of contact of the circle x^(2)+y^(2)+2x+2y+1=0 and x^(2)+y^(2)-2x+2y+1=0

Theorem : A necessary and sufficient condition for the quadratic equations a_(1)x^(2)+b_(1)x+c_(1)=0" and "a_(2)x^(2)+b_(2)x+c_(2)=0 to have a common roots is (c_(1)a_(2)-c_(2)a_(1))^(2)=(a_(1)b_(2)-a_(2)b_(1))(b_(1)c_(2)-b_(2)c_(1))

Show that the circles x^(2)+y^(2)+2ax+c=0 and x^(2)+y^(2)+2by+c=0 touch each other if 1/(a^(2))+1/(b^(2))=1/c

Show that the circles x^(2)+y^(2)-8x-2y+8=0 and x^(2)+y^(2)-2x+6y+6=0 touch each other and find the point of contact.