WMA11/01 Oct 2022, Question 9: Tangent and Normal Problems

Part (a) · 3 marks

Rewrite ½x² − 10x + 22 as a(x + b)² + c and state the constants a, b and c.

1. 1½x² − 10x + 22 = ½(x² − 20x) + 22
2. 2= ½[(x − 10)² − 100] + 22
3. 3= ½(x − 10)² − 50 + 22 = ½(x − 10)² − 28

Answer½(x − 10)² − 28

Part (b) · 2 marks

Use the completed-square form to write down the coordinates of the minimum point M.

1. 1Minimum when (x − 10)² = 0, so x = 10
2. 2y = −28, so M = (10, −28)

AnswerM(10, −28)

Part (ci) · 4 marks

Using the normal gradient −1/8, find the coordinates of the point P on C.

1. 1Normal gradient = −1/8, so tangent gradient = 8
2. 2dy/dx = x − 10 = 8 ⇒ x = 18
3. 3y = ½(18)² − 10(18) + 22 = 162 − 180 + 22 = 4
4. 4P = (18, 4)

AnswerP(18, 4)

Part (cii) · 2 marks

Substitute the coordinates of P into y = k − (1/8)x to find k.

1. 14 = k − 18/8
2. 2k = 4 + 9/4 = 25/4

Answerk = 25/4

Part (d) · 3 marks

Give the inequalities in x and y that describe the shaded region R bounded by l, C and the vertical line through M.

1. 1R is to the right of the line x = 10: x ≥ 10
2. 2R is below l: y ≤ 25/4 − (1/8)x
3. 3R is above C: y ≥ ½x² − 10x + 22

Answerx ≥ 10, y ≤ 25/4 − (1/8)x, y ≥ ½x² − 10x + 22

More past-paper questions on tangent and normal problems

• WMA11/01 Jan 2025 Q5 · 8 marks
• WMA11/01 Oct 2024 Q8 · 7 marks
• WMA11/01 Jan 2023 Q10 · 10 marks
• WMA11/01 Jun 2022 Q10 · 12 marks
• WMA11/01 Oct 2020 Q8 · 11 marks