If the circles $x^2 + y^2 + 2ax + b = 0 $ and $x^2 + y^2 + 2cx + b = 0 $ touch each other, then

Problem. If the circles $\large x^2 + y^2 + 2ax + b = 0 $ and $\large x^2 + y^2 + 2cx + b = 0 $touch each other, then

(A) b > 0

(B) b < 0

(D) none of these

Sol. Since the circles touch each other, equation of the common tangent is S₁– S₂= 0

⇒ x = 0.

Putting x = 0 in the equations of the circles, we get y²+ b = 0.

This equation should have equal roots ⇒ b = 0.

Hence (C) is the correct answer.