Hint
The radius is positive. Complete the square in x.
Worked solution
(a)
\(\left(x - 10\right)^{2} + \left(y - 3\right)^{2} = 49\) is the completed-square form.
Answer: \(\left( 10, \ 3\right)\); \(7\)
(b)
Substitution gives \(\displaystyle (x-h)^{2}=r^{2}\), with two signs.
Answer: \(\left\{\left( 3, \ 3\right), \left( 17, \ 3\right)\right\}\)
(c)
Move the square-completion constants to the other side.
\(49\)
Answer: \(49\)
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: Complete both coordinate squares and identify the positive radius.
- 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: The linear coefficients are −2h and −2k.
(b) · 2 marks
- 1 mark: Substitute the horizontal-line height and solve for both x-values.
- 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: The constant on one side is not automatically \(\displaystyle r^{2}\).
(c) · 2 marks
- 1 mark: Move the square-completion constants to the other side.
- 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: The linear coefficients are −2h and −2k.
Total: 6 marks.