Coordinate geometry — Different slope parameters

Question

Eliminate from , and write .

This practice question was inspired by Question 11(a) 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: convert between parametric and Cartesian forms carefully, noting the direction of the parameter..
State the final answer clearly in the form the question requested..

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

Verification: When , point is , which matches with : .

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