Hint
The specified point is on the circle. Take the negative reciprocal of \(\displaystyle \frac{v}{u}\).
Worked solution
(a)
The squared distance \(\displaystyle CP^{2}\) is 13, matching \(\displaystyle r^{2}\). The tangent is perpendicular to radius (2,3), so use its negative reciprocal gradient through P.
Answer: \(\displaystyle y = 15 - \frac{2 x}{3}\)
(b)
The normal is the radius line through C and P.
Answer: \(\displaystyle y - 5 = \frac{3 x}{2} - \frac{45}{2}\)
(c)
Sum the squares of the radius components.
\(13\)
Answer: \(13\)
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) · 3 marks
- 1 mark: Verify the point using squared distance to the centre.
- 1 mark: Use perpendicularity to its radius and pass the tangent through P.
- 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 the radius from the centre, not from the origin.
(b) · 2 marks
- 1 mark: Use the line joining the centre to P.
- 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 line through P is not automatically a tangent.
(c) · 2 marks
- 1 mark: Sum the squares of the radius components.
- 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: Use the radius from the centre, not from the origin.
Total: 7 marks.