Family of Circles Through Intersection of a Circle and a Line

Circle passing through the two points where circle S = 0 meets line L = 0. λ is chosen to impose one additional condition (e.g., passes through a given point).

Let circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$ meet line $L \equiv lx + my + n = 0$ at points $A$ and $B$.

Consider $S + \lambda L = 0$:

$$x^2 + y^2 + 2gx + 2fy + c + \lambda(lx + my + n) = 0$$ $$x^2 + y^2 + (2g + \lambda l)x + (2f + \lambda m)y + (c + \lambda n) = 0$$

The coefficients of $x^2$ and $y^2$ remain equal (both 1) and there is no $xy$ term — this is always a circle for any $\lambda$.

It passes through $A$ and $B$: Since $S(A) = 0$ and $L(A) = 0$, we have $S(A) + \lambda L(A) = 0$ for all $\lambda$. Same for $B$.

Usage: To find a specific member of this family, impose one condition. For example, "the circle also passes through $P(p, q)$" gives:

$$S(p, q) + \lambda L(p, q) = 0 \implies \lambda = -\frac{S(p,q)}{L(p,q)}$$

provided $P$ does not lie on $L$.

This family is especially useful when the two intersection points of the original circle and line are not easily computable.