We know that for the following lines to intersect the coordinate axes in concyclic points,the following condition must be met; $$ a_1a_2 = b_1b_2 $$ Where, $$ L_1 : a_1x+b_1y+c_1=0$$ $$ L_2 : a_2x+b_2y+c_2=0$$ Now, find the equation of the circle through them.
There are many methods that I can prove this but one of the method states that:
Let us consider the equation of pair of straight lines $ L_1L_2$, Now let $S$ be the equation of circle through the aforementioned concylic points then;
$$ S : L_1L_2 +\lambda xy $$
$$ S : a_1a_2x^2+b_1b_2y^2+(a_1b_2+a_2b_1)xy+(c_2a_1+c_1a_2)x+(c_2b_1+c_1b_2)y+c_1c_2 +\lambda xy=0$$
$$ S : a_1a_2x^2+b_1b_2y^2+(a_1b_2+a_2b_1+\lambda)xy+(c_2a_1+c_1a_2)x+(c_2b_1+c_1b_2)y+c_1c_2=0$$
Thus for $\lambda = -(a_1b_2+a_2b_1)$ we get the equation of circle. This method of solution seems very uninituitive to me, we just multiplied the equations of $L_1$ and $L_2$ and eliminated the $xy$ term and got the equation of circle. That makes no logical sense to me. Wasn’t it just a coincidence that through these steps We got the equation of circle. Let me illustrate my confusion by taking an example as well, Let $L_3L_4$ be the equation of pair of straight lines;
$$ L_3L_4 : x^2+y^2+2xy-1=0 $$
Now let us define $S’$ so that it is the equation of circle in the form of $L$. Thus;
$$ S’ : L_3L_4 +\alpha xy =0 $$
$$ S’ : x^2+y^2+(2+\alpha)xy-1=0 $$
Now we need to eliminate the $xy$ term to get the equation of the circle, For $\alpha = -2$ we get;
$$ S’ : x^2+y^2-1=0$$ as the equation of the circle.
But yet again, it does not make any physical sense so as to why the conversion works out the way it does?
Tl;DR What is the inituition/logical interpretation of conversion of equation of pair of straight lines to that of the circle by elimination of $xy$ term?
P.S. : What I mean to ask is that what is the logical interpretation of above. Like the logical explanation for the fact that just changing the eccentricity changes the way a conic looks, is derived out of the fact that logically, changing eccentricity is same as changing the intersection locus of plane and cone. Sorry for the trouble.
Source:Wikipedia
The intuition behind defining $S$ as $L_1L_2+\lambda xy=0$ was to make sure that we get:
$1.$ A $2$-degree equation
$2.$ An equation which satisfies all $4$ points where lines intersect axes
You will see that no other combination of terms satisfies both of these criteria. However, if this does not seem intuitive to you, you could always first take any of the three points, write the equation of the circle through them, and make sure the $4^{th}$ point passes through them. The same condition is obtained.
Edit: A unique circle passes through $3$ given points. Hence, since all $4$ points satisfy the given equation, this proves that $S$ can only be the equation of the circle that we are looking for, and not any other circle.