# Thread: what is the line obtained from the equation of 2 intersecting circles?

1. ## what is the line obtained from the equation of 2 intersecting circles?

Hi, I am struggling with this problem . I do not know the properties of circles so well, so I gave the variables some values and drew, but I still don't get anything :-/. It would also be great if you could tell me how you know (what rule, etc.) you have used to answer the question. Thanks!

The problem says:

Let $ax^2+ay^2+kx+ly+m=0$, $bx^2+by^2+px+qy+r=0$ be two intersecting circles. Deviding the first equation by a and the second by b and subtracting, we obtain the equation of a line:

$(k/a-p/b)x+(l/a-q/b)y+(m/a-r/b)=0$.

What is this line?

2. Is it the line connecting the 2 points of intersection between the circles?

3. Is that a line?
Does it exist for x^2 + y^2 = 1 vs. (x-5)^2 + y^2 = 1
Under what conditions on k, a, p, b, and q does it not exist?

4. I am so lost... I drew and did not get a line connecting the two points of intersection, just a random line. Maybe I did it wrong, but I did it twice and got the same thing . Maybe there is a mistake there because it does not make much sense :-/

5. For starters, the intersection is 0 points, 1 point, 2 points, or infinitely many points in a circular locus. If you are doing something that leads to a linear equation, that seems kind of magic, doesn't it? Why would you believe that would work? It might. Feel free to prove it. Note: It does give the equation of the line, but it does NOT give the points of intersection.

Second, not everything that looks like that is a circle. There must be relationships between the parameters. If you enforce the relationships, you will see that it is all quite a bit simpler. Expand $(x-a)^{2} + (y-b)^{2} = r^{2}$, and put it in that standard form. You will see the necessary relationships.

Have you a particular numerical example that you find vexing?

Exploration: If there is no intersection, what does it produce? It's a line perpendicular to what?

6. Originally Posted by juanma101285
Hi, I am struggling with this problem . I do not know the properties of circles so well, so I gave the variables some values and drew, but I still don't get anything :-/. It would also be great if you could tell me how you know (what rule, etc.) you have used to answer the question. Thanks!

The problem says:

Let $ax^2+ay^2+kx+ly+m=0$, $bx^2+by^2+px+qy+r=0$ be two intersecting circles. Deviding the first equation by a and the second by b and subtracting, we obtain the equation of a line:

$(k/a-p/b)x+(l/a-q/b)y+(m/a-r/b)=0$.

What is this line?