2023HSCShort answer

Q14 (b) (i)

1 mark

Question

Consider the hyperbola $y=\dfrac{1}{x}$ and the circle $(x-c)^2+y^2=c^2$ , where $c$ is a constant.

(i) Show that the $x$ -coordinates of any points of intersection of the hyperbola and circle are zeros of the polynomial $P(x)=x^4-2cx^3+1$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	1

Sample answer:

The points of intersection of the hyperbola and the circle satisfy

\begin{cases}y=\frac1x &(1)\\(x-c)^2+y^2=c^2 &(2)\end{cases}

Substitute (1) into (2):

\begin{aligned} (x-c)^2+\frac1{x^2}&=c^2\\ x^2(x^2-2cx+c^2)+1&=c^2x^2\\ x^4-2cx^3+1&=0 \end{aligned}

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Syllabus alignment

Functions

PolynomialsUse polynomial division, the factor and remainder theorems, zeros and graphs to solve polynomial problems.

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