In the chapter Circles, under the topic Family of Circles, I came across the following statement:
Family of circles circumscribing a triangle whose sides are given by $L_1=0$, $L_2=0$ and $L_3=0$ is given by $L_1L_2+\lambda L_2L_3+\mu L_3L_1=0$ provided coefficient of $xy=0$ and coefficient of $x^2$ and $y^2$ are equal.
We know that given a triangle, we have three fixed points and there is only one circle which passes through all the three points at the same time. Here, we have been given three lines, which give us the vertices of the triangle. We will be having only one circle which passes through all the three points.
Then why is the book stating a family of circles? How can two or more circles pass through three fixed points at the same point? Is the above statement correct? If yes, kindly give some explanation on how to arrive at this result, as my book gives no proof for this.
I am unable to find any relevant resources regarding this on the internet.
Best Answer
First, observe that the given family is a family of conics (being a quadratic form =0). At a vertex of the triangle you have two of the linear functions $L_i$ equal zero and then the point belongs to the conic (say if $L_1=L_2=0$ at the vertex the left hand side of your equation =0). So the conics of your family circumscribe the triangle. In order for the conic to be a circle you need $x^2$ and $y^2$ to have equal coefficients and $xy$ to be absent. When you impose those two conditions you are left with just one circle (two equations with two unknowns $\lambda$ and $\mu$) which is the circumscribed circle.