Compare centre distance with the sum or difference of the radii.
Notes and key methods
For positive radii \(\displaystyle r_{1},r_{2}\), external contact has centre distance \(\displaystyle r_{1}+r_{2}\). Internal contact has distance \(\displaystyle |r_{1}-r_{2}|\), provided the circles are distinct. Both contact points lie on the line of centres.
Two intersections occur when \(\displaystyle |r_{1}-r_{2}|<d<r_{1}+r_{2}\). Distances outside these bounds give separate circles or one strictly inside the other. Coincident equal circles require a separate case.
Check the conditions
Both radii are positive and the circles are distinct.
Identify external versus internal contact before using a radius sum.
Three worked examples
Example 1 · Touching circles
Circle 1 has centre \(\left( 5, \ 1\right)\) and radius 7. Circle 2 has centre \(\left( 9, \ 1\right)\) and radius 3.
Determine the type of contact and justify it.
Find the contact point and verify it on both circles.
Find the contact point y-coordinate.
Find the constant term in the expanded second-circle equation.
Show the reasoning
(a)
The centre distance is 4, which equals \(\displaystyle r_{1}-r_{2}\), with r₁>r₂.
Answer: Internal contact
(b)
Its distances from \(\displaystyle C_{1}\) and \(\displaystyle C_{2}\) are 7 and 3, respectively; the point is on their line of centres.
Answer: \(\left( 12, \ 1\right)\)
(c)
Use the line of centres.
\(1\)
Answer: \(1\).
(d)
Expand both coordinate squares and move \(\displaystyle q^{2}\) to the left.
\(73\)
Answer: \(73\).
Example 2 · Touching circles
Circle 1 has centre \(\left( 6, \ 2\right)\) and radius 9. Circle 2 has centre \(\left( 19, \ 2\right)\) and radius 4.
Determine the type of contact and justify it.
Find the contact point and verify it on both circles.
Find the contact point y-coordinate.
Find the constant term in the expanded second-circle equation.
Show the reasoning
(a)
The centre distance is 13, which equals \(\displaystyle r_{1}+r_{2}\).
Answer: External contact
(b)
Its distances from \(\displaystyle C_{1}\) and \(\displaystyle C_{2}\) are 9 and 4, respectively; the point is on their line of centres.
Answer: \(\left( 15, \ 2\right)\)
(c)
Use the line of centres.
\(2\)
Answer: \(2\).
(d)
Expand both coordinate squares and move \(\displaystyle q^{2}\) to the left.
\(349\)
Answer: \(349\).
Example 3 · Touching circles
Circle 1 has centre \(\left( 7, \ 3\right)\) and radius 5. Circle 2 has centre \(\left( 10, \ 3\right)\) and radius 2.
Determine the type of contact and justify it.
Find the contact point and verify it on both circles.
Find the contact point y-coordinate.
Find the constant term in the expanded second-circle equation.
Show the reasoning
(a)
The centre distance is 3, which equals \(\displaystyle r_{1}-r_{2}\), with r₁>r₂.
Answer: Internal contact
(b)
Its distances from \(\displaystyle C_{1}\) and \(\displaystyle C_{2}\) are 5 and 2, respectively; the point is on their line of centres.
Answer: \(\left( 12, \ 3\right)\)
(c)
Use the line of centres.
\(3\)
Answer: \(3\).
(d)
Expand both coordinate squares and move \(\displaystyle q^{2}\) to the left.
\(105\)
Answer: \(105\).
Common mistakes
Internal contact uses a radius difference.
Concentric equal circles are not a one-point tangency case.