Intersection of two circles

The intersection of two circles is a point lying on both circles at once. Here we look at how the circle $x^2 + y^2 = 25$, centered at the origin $(0, 0)$ with radius $5$, meets the circle $(x - 8)^2 + y^2 = 25$, centered at $(8, 0)$ with radius $5$.

At an intersection both equations hold at once. Subtracting one from the other cancels the $x^2$ and $y^2$ terms and leaves a linear equation.

Expanding gives $x^2 - (x^2 - 16x + 64) = 16x - 64$, so $16x - 64 = 0$, that is $x = 4$. Substituting into $x^2 + y^2 = 25$ gives $16 + y^2 = 25$, so $y^2 = 9$ and $y = 3, -3$. The intersection points are $(4, 3)$ and $(4, -3)$.

The linear equation $x = 4$ that came out of the subtraction is the line through the two intersection points (the line containing the common chord). To find where two circles meet, the standard method is to subtract first to get this line, then substitute it into a circle equation.

How the circles meet is decided by the distance between centers $d$ compared with the two radii $r_1, r_2$.

• $|r_1 - r_2| < d < r_1 + r_2$: they cross at 2 points
• $d = r_1 + r_2$ (external) or $d = |r_1 - r_2|$ (internal): they touch at 1 point
• $d > r_1 + r_2$ (apart) or $d < |r_1 - r_2|$ (one inside the other): they do not meet

Here $d = 8$, $r_1 + r_2 = 10$ and $|r_1 - r_2| = 0$, so $0 < 8 < 10$ and the circles cross at 2 points, as we found. The large dots on the graph are those two intersection points.