Vectors — Area formula then double-angle circles

Question

(i) Using a perpendicular vector of equal magnitude, show that the area of is .

(ii) Hence, with and , show the area of is (up to an equivalent non-negative form), and find maximising the area.

This practice question was inspired by Question 14(c) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.(i) Same perpendicular-vector argument: area .
Simplify the previous line carefully, keeping exact values where possible.(ii) Expand and simplify to .
Finish the calculation, then check that the result meets the question’s conditions.Let . Then .
State the final answer clearly in the form the question requested.Stationary points with give in as the area maximisers (by the first-derivative sign chart / evenness).

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Dependent: area identity from (i) feeds the maximisation in (ii).

Any mark labels are a TestMum study aid, not an official NESA marking allocation.