Intersection of two circles
You may be asked to show that two circles are touching, and say whether they're touching internally or externally.
To do this, you need to work out the radius and the centre of each circle.
If the sum of the radii and the distance between the centres are equal, then the circles touch externally.
If the difference between the radii and the distance between the centres are equal, then the circles touch internally.
Determining whether two circles touch each other
Two circles will touch if the distance between their centres, \(d\), is equal to the sum of their radii, or the difference between their radii.
Two circles will intersect at two points when \({r_1} - {r_2}\textless d\textless{r_1} + {r_2}\)
The centre of one circle will lie on the other circle when \(d = {r_1}\) or \(d = {r_2}\).
Two circles are concentric when \(d = 0\).
Example 1
Do the circles \({(x - 1)^2} + {(y - 1)^2} = 9\) and \({(x - 5)^2} + {(y - 4)^2} = 4\) touch and if so, in what way?
Solution
\({C_1} = (1,1)\) and \({C_2} = (5,4)\)
\({C_1}{C_2} = \sqrt {{4^2} + {3^2}} = 5\)
\({r_1} = 3\) and \({r_2} = 2\)
\(\Rightarrow {C_1}{C_2} = {r_1} + {r_2}\)
The circles touch externally.
Example 2
Do the circles \({(x - 2)^2} + {(y - 1)^2} = 5\) and \({(x - 4)^2} + {(y - 5)^2} = 45\) touch and if so, in what way?
Solution
\({C_1} = (2,1)\) and \({C_2} = (4,5)\)
\({C_1}{C_2} = \sqrt {{2^2} + {4^2}} = \sqrt {20} = 2\sqrt 5\)
\({r_1} = \sqrt 5\) and \({r_2} = \sqrt {45} = 3\sqrt 5\)
\(\Rightarrow {C_1}{C_2} = {r_2} - {r_1}\)
The circles touch internally.
