The equation of a family of circles passing through two given points

Question

We have to write the equation of a family of circles passing through two given points $(x_1,y_1)$ & $(x_2, y_2)$

From these points , I only know one circle that is

$$(x_1-x)(x_2-x)+(y-y_1)(y-y_2)=0$$

But now how Can I write other equations .

Answer 1

Family of circle passing through two given points $A(x_1, y_1)$ and $B(x_2, y_2)$.

For more details see this.

Answer 2

We know that $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) $ is the equation of the circle having the two points as it's diameter and that $\lambda \begin {vmatrix} x & y & 1\\ x_1 & y_1 & 1\\ x_2 & y_2 & 1\end {vmatrix} $ is the equation of a line passing through the two points. Thus, the required family of circles is $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) +\lambda \begin {vmatrix} x & y & 1\\ x_1 & y_1 & 1\\ x_2 & y_2 & 1\end {vmatrix} =0 $$

Also see here. Hope it helps.

Answer 3

Here is an understanding of what it means to change $\lambda$ in the equation: $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) +\lambda \begin {vmatrix} x & y & 1\\ x_1 & y_1 & 1\\ x_2 & y_2 & 1\end {vmatrix} =0 $$

Fixing the point as $(x_1,y_1)=0$ and $(x_2,y_2)=(2,1)$, find my graph here. Here is a gif of the above:

Explanation: When $\lambda=0$, at that point the circle is the one having diameter as the line segment from $(x_1,y_1)$ and $(x_2,y_2)$. As we vary $\lambda$ we get a family of circles still passing through these two points on the line (and no other one the line passing through them) but this time this line segment is chord of the circle.