### Variation Point under a parabola, long division of polynomials

Question Sample Titled 'Variation Point under a parabola, long division of polynomials'

 (a) Find the value of ${s}$ such that ${x}-{3}$ is a factor of ${s}{x}^{{3}}-{68}{x}^{{2}}+{111}{x}-{18}$ . (2 marks) (b) The figure shows the graph of ${y}={33}{x}^{{2}}-{204}{x}+{333}$ . ${P}$ is a variable point on the graph in the first quadrant. ${C}$ and ${V}$ are the feet of the perpendiculars from ${P}$ to the ${x}$-axis and the ${y}$-axis respectively. (5 marks)

${x}$${y}$${C}$${P}$${V}$${O}$${y}={33}{x}^{{2}}-{204}{x}+{333}$

 (i) Let ${\left({v},{0}\right)}$ be the coordinates of ${C}$. Express the area of the rectangle ${O}{C}{P}{V}$ in terms of ${v}$.  (ii) Are there three different positions of ${P}$ such that the area of the rectangle ${O}{C}{P}{V}$ is ${54}$ ? Explain your answer.

 (a) ${s}{\left({3}\right)}^{{3}}-{68}{\left({3}\right)}^{{2}}+{111}{\left({3}\right)}-{18}$ $={0}$ 1M ${27}{s}$ $={297}$ ${s}$ $={11}$ 1A (bi) The area of the rectangle ${O}{C}{P}{V}$ $={v}{\left({33}{v}^{{2}}-{204}{v}+{333}\right)}$ 1A  $={33}{v}^{{3}}-{204}{v}^{{2}}+{333}{v}$ (bii) Note that the area of the rectangle ${O}{C}{P}{V}$ is ${54}$ . Thus, we have ${33}{v}^{{3}}-{204}{v}^{{2}}+{333}{v}$ $={54}$ 1M ${11}{v}^{{3}}-{68}{v}^{{2}}+{111}{v}-{18}$ $={0}$ ${\left({v}-{3}\right)}{\left({11}{v}^{{2}}-{35}{v}+{6}\right)}$ $={0}$ 1M ${\left({v}-{3}\right)}^{{2}}{\left({11}{v}-{2}\right)}$ $={0}$ 1A ${v}$ $={3}{\quad\text{or}\quad}{v}=\dfrac{{2}}{{11}}$ So, there are only two (no three) different positions of ${P}$ such that the area of the rectangle ${O}{C}{P}{V}$ is ${54}$ . 1A f.t.

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