How to find the common chord of two circles.

Question

"Two circles, each of radius 5 units, have centers at the origin and (7,7), respectively. What is the y-intercept of the line that contains their common chord?"

I'm in 7th grade and I got this problem in Mathcounts. But my teacher didn't explain it to us because she didn't want to confuse us, as we don't learn about chords until tenth grade. However, I am an over-achiever and she said that she expected no one in middle school to figure it out, which of course made me want to know how to solve it, and my mind will not be at rest until I figure it out. I tried searching it up, but all that appeared was, "we will subtract the equation (ii) from the equation (i). ⇒ 2x + 12y + 27 = 0, which is the required equation. The slope of the common chord 2x + 12y + 27 = 0 is (m1) = -16. Centre of the circle x2 + y2 - 4x - 2y - 31 = 0 is (2, 1)," which to me is super confusing, I don't even know what to do with those equations. So my point is, can you explain to me how to find the common chord of two circles?

Answer 1

When two circles overlap, the common chord is the line that connects their points of intersection.

The equations of circles are in the form (x-h)²+(y-k)²=r² where r is the radius and (h,k) is the center. once you have the 2 equations for each circle, you can set the left sides equal to each other since they both have a radius of 5. This will give you a linear equation that represents the line where the distance between the centers of each circle and the line are equal. This includes the intersect points since the distance to each intersect from the center is 5, so this linear equation is also the equation of the chord. Once you put this equation into slope intercept form you will have the y-intercept.