Polynomial roots — Hyperbola $y=2/x$

Question

Consider the hyperbola and the circle , where is a positive constant.

(i) Show that the -coordinates of any points of intersection are zeros of .

(ii) Hence, by solving , find the exact value of such that the hyperbola and circle intersect at only one point.

This practice question was inspired by Question 14(b) 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: translate the requested symmetric expression into sums and products of roots from the coefficients.(i) Substitute : . Multiply by : .
Simplify the previous line carefully, keeping exact values where possible.Expand: .
Simplify the previous line carefully, keeping exact values where possible.(ii) or . .
Finish the calculation, then check that the result meets the question’s conditions.: .
State the final answer clearly in the form the question requested. (since ).

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

Verification: Full (i)–(ii): eliminate , then double-root condition for single intersection.

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