The circles x^2+y^2+2 a x+ c =0 and x^2+y^2+2 ~b y+ c =0 touch each other externally, if

Step-by-step solution

aligned & x^2+y^2+2 a x+ c =0 & x^2+2 a x+ a ^2+y^2= a ^2- c & (x+ a )^2+y^2= ( a ^2- c )^2 aligned i.e., it is a circle with centre (-a, 0) and radius a ^2- c Simiłarly, aligned & x^2+y^2+2 ~b y+ c =0 & x^2+(y+ b )^2= ( b ^2- c )^2 aligned i.e., it is a circle with centre (0,-b) and radius =t b^2-c If circles touch externally, then we get Sum of radii = Distance between centres aligned & a^2-c + b^2-c = a^2+b^2 & a^2-c+b^2-c+2 a^2-c b^2-c =a^2+b^2 & (a^2-c ) (b^2-c )=c^2 & a^2 b^2-c b^2-c a^2+c^2=c^2 & 1 a^2 + 1 b