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?
Best Answer
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 $P$.