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2023 HSC Mathematics Extension 1

Question 14(b):
Polynomial roots

4 markspolynomial rootsdependency-safe group

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

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

(ii) By considering the given graphs of for and , or otherwise, find the exact value of such that the hyperbola and circle intersect at only one point.

The supplied graphs of y = x^4 - 2cx^3 + 1 for c = 0.8 and c = 1.
The supplied graphs of y = x^4 - 2cx^3 + 1 for c = 0.8 and c = 1.

How to recognise this question

Question type: polynomial roots

  • The command and notation point to polynomial roots.
  • The word “hence” means the later part is intended to reuse the earlier result.

How to handle it: translate the requested symmetric expression into sums and products of roots from the coefficients.

Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) Substitute into the circle: .
  2. Multiply through by : .
  3. Expand: .
  4. NESA (i): 1 mark.
  5. (ii) Single intersection (with ) corresponds to a repeated root of : solve and simultaneously.
  6. or .
  7. , so the double-root candidate is .
  8. : .
  9. .
  10. (since ).
  11. NESA (ii): 3 marks; 2 for -coordinate of double root in terms of ; 1 for recognising double-root / tangency condition.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 4 total marks 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

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.

See the step-by-step answer guide →

Question 2

Consider and with .

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

(ii) Hence find the exact for which there is only one intersection point.

See the step-by-step answer guide →

Question 3

Consider and , .

(i) Show that intersection -coordinates satisfy .

(ii) Hence, by imposing a repeated root of , find the exact value of for a single intersection point.

See the step-by-step answer guide →