2021 HSC Mathematics Extension 1
Question 8:
Coordinate geometry
The diagram shows a semicircle of radius with diameter endpoints and and lowest point (centre ).
Which pair of parametric equations represents the semicircle shown?
How to recognise this question
Question type: coordinate geometry
- The command and notation point to coordinate geometry.
- The requested response is a multiple choice.
How to handle it: convert between parametric and Cartesian forms carefully, noting the direction of the parameter.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Centre , radius . The diagram is the lower semicircle: diameter on from to , bulging downward to .
- Option C: at , ; at , ; at , . These are exactly the three key points of the lower semicircle.
- Option A places the semicircle above the diameter (upper half). Option B is a right-to-left upper/lower mix with ranging on that -interval but starting at and ending at through (upper). Option D traces differently and does not match the labelled points simultaneously.
- Answer: C. (NESA multiple-choice key: C.)
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Which pair parametrises the upper semicircle centre , radius , diameter from to ?
Question 2
Which pair parametrises the lower semicircle centre , radius , diameter on the -axis from to ?
Question 3
For , , , which point is the bottom of the arc?