Hint
Use the derivative for the parabola, distance for the circle, and substitution for a separate tangency check.
Worked solution
(a)
The derivative is 2x−2, so its value at \(\displaystyle x=2\) is 2. The tangent through (2,2) is \(\displaystyle y-2=2(x-2)\), hence \(\displaystyle y=2x-2\).
Answer: \(y=2x-2\).
(b)
The squared radius is \(\displaystyle CP^{2}=(2-4)^{2}+(2-1)^{2}=5\). With centre (4,1), the equation is \(\displaystyle (x-4)^{2}+(y-1)^{2}=5\).
Answer: \((x-4)^2+(y-1)^2=5\).
(c)
Substituting \(\displaystyle y=2x-2\) gives \(\displaystyle (x-4)^{2}+(2x-3)^{2}=5\). Expanding reduces it to \(\displaystyle 5x^{2}-20x+20=5(x-2)^{2}=0\). Thus \(\displaystyle x=2\) is the only, repeated root and the line gives \(\displaystyle y=2\). The unique circle contact is P, so the line is tangent there.
Answer: The equation reduces to \(5(x-2)^2=0\), giving only P.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Find the derivative 2x−2.
- 1 mark: Use slope 2 and the point P.
- 1 mark: Give the correct tangent equation.
Alternative methods: A first-principles derivative is accepted.
Common error: Using the ordinate as a slope without differentiating the curve.
(b) · 2 marks
- 1 mark: Find the squared radius 5.
- 1 mark: Give the correct circle equation.
Alternative methods: An equivalent expanded circle equation is accepted.
Common error: Using 5 as the radius and squaring it again.
(c) · 3 marks
- 1 mark: Substitute the correct line into the circle.
- 1 mark: Obtain the repeated-root equation \(\displaystyle 5(x-2)^{2}=0\).
- 1 mark: Identify the unique point P and conclude tangency.
Alternative methods: An equivalent substitution followed by a zero discriminant is accepted.
Common error: Asserting that the parabola tangent is automatically a circle tangent without checking the circle.
Total: 8 marks.