Family of Circles Touching a Circle and a Line

Question

My book says -

The equation of the family of circles touching the circle $S = 0$ and the line $L = 0$ at their point of contact $P$ is -
                                                                   $S + \lambda L = 0$     where $\lambda $ is a parameter.

In the above equation, $S=0$ and $L=0$ both satisfy the coordinates of $P$. Hence, $S + \lambda L = 0$ also satisfies the coordinates of $P$. I am not able to find any condition that makes the circle $S + \lambda L = 0$ only touch the circle $S=0$ and the line $L=0$.

I feel this is the equation of family of circles passing through point of contact $P$ rather than the equation of family of circles touching the circle and the line at $P$ exclusively.

What am I missing here?

Answer 1

if $(S+ \lambda L = 0)$ intersected the line or the circle at any other point, it would also intersect the other object there, so the initial circle and line would have two intersection points. Since this is not the case, $P$ is the only intersection of $(S+ \lambda L=0)$ with either the circle or the line, so it is tangent to them at $P$.

Answer 2

Let $C = S+\lambda L$

$C = 0$ is a second degree equation in both $x$ and $y$ and since it is tangent to both $S = 0$ and $L = 0$, it has two coincident roots with both of the curves. Note that $C=0$ cannot intersect either of the two again as second degree equation cannot have more than two roots!