Hint
Both radii are positive and the circles are distinct. The centres have the same y-coordinate.
Worked solution
(a)
The centre distance is 5, 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 9 and 4, respectively; the point is on their line of centres.
Answer: \(\left( 18, \ 1\right)\)
(c)
Use the line of centres.
\(1\)
Answer: \(1\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Compare the centre distance with the radius sum and absolute radius difference.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Internal contact uses a radius difference.
(b) · 2 marks
- 1 mark: Locate the contact on the line of centres and verify both distances.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Concentric equal circles are not a one-point tangency case.
(c) · 2 marks
- 1 mark: Use the line of centres.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Internal contact uses a radius difference.
Total: 6 marks.