Hint
Both coordinates must satisfy both equations. Substitute the line, then take both square-root branches.
Worked solution
(a)
Substitution leaves \(\displaystyle (x-9)^{2}=16\), so \(\displaystyle x=9\pm 4\).
Answer: \(\left\{\left( 5, \ 8\right), \left( 13, \ 8\right)\right\}\)
(b)
\(8\) agrees with the coordinate difference.
Answer: \(8\)
(c)
Subtract the two intersection x-coordinates.
\(8\)
Answer: \(8\)
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: Substitute the line in the circle and retain both coordinate pairs.
- 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: A repeated x-root is one contact point.
(b) · 2 marks
- 1 mark: Use the perpendicular from the centre and Pythagoras to find the half-chord, then double it.
- 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: Use a square root with both signs for the two intersections.
(c) · 2 marks
- 1 mark: Subtract the two intersection x-coordinates.
- 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: A repeated x-root is one contact point.
Total: 6 marks.